Integrals of $\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} f(k)\text{d}k$

Question

Consider a type of integrals $$ \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} f(k)\text{d}k $$ where $K=K(k),K^\prime=K(\sqrt{1-k^2})$ are complete elliptic integrals, and $k$ is an elliptic modulus. $f(k)$ will be chosen if it satisfies some properties. And meanwhile their values can be obtained in brief forms(expressed by Dirichlet $L$-series in most cases). Several cases have been considered in Question.1, Question.2. The paper gave out numerous and rich examples, e.g. $$ \left ( \frac{2}{\pi} \right )^2 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} K(k)\text{d}k=2^{s+2}\pi^{-s}\Gamma(s)\lambda(s)\beta(s-2). $$ To start, I should list the $L$-series to be used. $$ \beta(s)=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^s},\eta(s)=(1-2^{1-s})\zeta(s),\lambda(s)=(1-2^{-s})\zeta(s),\\ $$ $$ L_8(s)=\sum_{n=0}^{\infty}\left ( \frac{1}{(8n+1)^s} - \frac{1}{(8n+3)^s}- \frac{1}{(8n+5)^s}+ \frac{1}{(8n+7)^s}\right ),\\L_{-8}(s)=\sum_{n=0}^{\infty}\left ( \frac{1}{(8n+1)^s}+ \frac{1}{(8n+3)^s}- \frac{1}{(8n+5)^s}-\frac{1}{(8n+7)^s}\right ),\\ L_{-20}(s) =\sum_{n=1}^{\infty} \left ( \frac{-20}{n} \right ) \frac{1}{n^s}. $$ $(\frac{m}{n})$ is the Kronecker symbol.
The main instructions are the same as what did in the paper. Consider $$ \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} g(k)\text{d}\left ( \frac{K^\prime}{K} \right ). $$ By making a substitution $x=K^\prime/K$, we have $$ \int_{0}^{\infty}x^{s-1}g\left ( \frac{\theta_2(q)^2}{\theta_3(q)^2} \right ) \text{d}x. $$

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It's clear that $$ \frac{2K}{\pi} =1+2\sum_{n=1}^{\infty} \frac{1}{\cosh(\pi nx)},x=K^\prime/K. $$ Taking mellin transforms both sides. Suppose for $s$ large sufficiently, we have \begin{aligned} \int_{0}^{\infty}x^{s-1}\left ( \frac{2K}{\pi} -1 \right )\text{d}x & = 2\int_{0}^{\infty} \sum_{n = 1}^{\infty} \frac{x^{s-1}}{\cosh(\pi n x)} \text{d} x\\ &=2\sum_{n=1}^{\infty}\frac{1}{(\pi n)^s} \int_{0}^{\infty} \frac{x^{s-1}}{\cosh(x)} \text{d}x\\ &=4\pi^{-s}\Gamma(s)\zeta(s)\beta(s). \end{aligned}

• Now consider series of functions, $$ f(z)=\operatorname{ns}(z,k^\prime) \left(\frac{\mathrm{d}^{2n}}{\mathrm{d}y^{2n}} \frac{1}{\cosh(y)} \right)\Big|_{y\rightarrow\frac{\pi z}{2K} }. $$ Follow same steps we obtain: \begin{aligned} &\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{(1-k^2)K(k)} \text{d}k =2^{s+2}\pi^{-s}\Gamma(s)\lambda(s)\beta(s),\\ &\left ( \frac{2}{\pi} \right )^2 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} K(k)\text{d}k=2^{s+2}\pi^{-s}\Gamma(s)\lambda(s)\beta(s-2),\\ &\left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} (1-5k^2)K(k)^3\text{d}k=2^{s+2}\pi^{-s}\Gamma(s)\lambda(s)\beta(s-4),\\ &\left ( \frac{2}{\pi} \right )^6 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} (1-46k^2+61k^4)K(k)^5\text{d}k=2^{s+2}\pi^{-s}\Gamma(s)\lambda(s)\beta(s-6),\\ &\left ( \frac{2}{\pi} \right )^8 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} (1-411k^2+1731k^4-1385k^6)K(k)^7\text{d}k=2^{s+2}\pi^{-s}\Gamma(s)\lambda(s)\beta(s-8),\\ &\left ( \frac{2}{\pi} \right )^{10} \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} (1-3692k^2+41838k^4-88412k^6+50521k^8)K(k)^9\text{d}k=2^{s+2}\pi^{-s}\Gamma(s)\lambda(s)\beta(s-10) \end{aligned} and etc.
• Consider $$ f(z)=\operatorname{nc}(z,k^\prime) \left(\frac{\mathrm{d}^{2n+1}}{\mathrm{d}y^{2n+1}} \frac{1}{\cosh(y)} \right)\Big|_{y\rightarrow\frac{\pi z}{2K} }, $$ which generates $\beta(s)\beta(s-2n-1)$: $$ \left ( \frac{2}{\pi} \right ) \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{\sqrt{1-k^2}}\text{d}k=2^{s+2}\pi^{-s}\Gamma(s)\beta(s)\beta(s-1), $$ $$ \left ( \frac{2}{\pi} \right )^3 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{(1-2k^2)K(k)^2}{\sqrt{1-k^2}}\text{d}k=2^{s+2}\pi^{-s}\Gamma(s)\beta(s)\beta(s-3), $$ $$ \left ( \frac{2}{\pi} \right )^{11} \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{P_{11}(k)K(k)^{10}}{\sqrt{1-k^2}}\text{d}k=2^{s+2}\pi^{-s}\Gamma(s)\beta(s)\beta(s-11), $$ where $P_{11}(k)=1-11074k^2+210112k^4-729728k^6+884480k^8-353792k^{10}$.
• Consider $$ f(z)=\operatorname{sc}(z,k^\prime) \left(\frac{\mathrm{d}^{2n}}{\mathrm{d}y^{2n}} \frac{\cosh(y)}{\cosh(2y)} \right)\Big|_{y\rightarrow\frac{\pi z}{2K} }, $$ which generates $\lambda(s)L_{-8}(s-2n)$. We have $$ \frac{2\sqrt{2} }{\pi^2} \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\left(1+k^2+\sqrt{1-k^2}\right) }{\sqrt{1-k^2} ( 1+\sqrt{1-k^2} )^{3/2} }K(k) \text{d}k =2^{s+1}\pi^{-s}\Gamma(s)\lambda(s)L_{-8}(s-2) $$ Or can be written in a pleasant form: $$ \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{(1+3k)K(k)}{\sqrt{k(1+k)} } \text{d}k =2^{2s-1}\pi^{2-s}\Gamma(s)\lambda(s)L_{-8}(s-2). $$
• Consider $$ f(z)=\operatorname{nc}(z,k^\prime) \left(\frac{\mathrm{d}^{2n+1}}{\mathrm{d}y^{2n+1}} \frac{\cosh(y)}{\cosh(2y)} \right)\Big|_{y\rightarrow\frac{\pi z}{2K} }, $$ which generates $\beta(s)L_{-8}(s-2n-1)$. We have $$ \frac{4\sqrt{2} }{\pi^3} \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\left(1-2k^2+\sqrt{1-k^2}(1+4k^2)\right) }{(1-k^2)^{3/4} \left ( 1+\sqrt{1-k^2} \right )^{3/2} }K(k)^2\text{d}k =2^{s+1}\pi^{-s}\Gamma(s)\beta(s)L_{-8}(s-3) $$ Some other examples are considered: $$ \frac{\sqrt{2} }{\pi} \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{\sqrt{1-k^2}\sqrt{1+\sqrt{1-k^2} } } \text{d}k =2^{s+1}\pi^{-s}\Gamma(s)\lambda(s)L_8(s-1), $$ $$\frac{4\sqrt{2} }{\pi^3} \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\left ( 1+k^2+\frac{1-5k^2}{\sqrt{1-k^2} } \right ) }{(1+\sqrt{1-k^2})^{3/2} }K(k)^2\text{d}k =2^{s+1}\pi^{-s}\Gamma(s)\lambda(s)L_8(s-3). $$

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Question: How to discover more generalizations, or how to find a method extensively calculating these integrals?

Answer 1

More Examples: Consider $$ f(z)=\left[\operatorname{nc}\left(z-\frac{K^\prime}{2},k^\prime\right)-\operatorname{nc}\left(z+\frac{K^\prime}{2},k^\prime\right)\right] \left(\frac{\mathrm{d}^{2n}}{\mathrm{d}y^{2n}} \frac{\cosh(y)}{\cosh(2y)} \right)\Big|_{y\rightarrow\frac{\pi z}{2K} }, $$ we have $$ \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{\sqrt{k}(1-k^2)^{3/4}K(k)}\text{d}k =2^{2s+1}\pi^{-s}\Gamma(s)L_8(s)L_{-8}(s),\\ \color{green}{\left(\frac2\pi\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{(1-2k^2+2k\sqrt{1-k^2})}{\sqrt{k}(1-k^2)^{3/4}}K(k)\text{d}k =2^{2s+1}\pi^{-s}\Gamma(s)L_8(s)L_{-8}(s-2).} $$ Consider $$ f(z)=\left[\operatorname{nc}\left(z-\frac{K^\prime}{2},k^\prime\right)+\operatorname{nc}\left(z+\frac{K^\prime}{2},k^\prime\right)\right] \left(\frac{\mathrm{d}^{2n}}{\mathrm{d}y^{2n}} \frac{\sinh(y)}{\cosh(2y)} \right)\Big|_{y\rightarrow\frac{\pi z}{2K} }, $$ we have $$ \color{green}{\left(\frac2\pi\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{(1-2k^2-2k\sqrt{1-k^2})}{\sqrt{k}(1-k^2)^{3/4}}K(k)\text{d}k =2^{2s+1}\pi^{-s}\Gamma(s)L_{-8}(s)L_{8}(s-2).} $$ From a combination of the green integrals, we conclude \begin{aligned} &\color{purple}{\left(\frac2\pi\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{(1-2k^2)K(k)}{\sqrt{k}(1-k^2)^{3/4}}\text{d}k =2^{2s}\pi^{-s}\Gamma(s)[L_8(s)L_{-8}(s-2)+L_{-8}(s)L_{8}(s-2)]},\\ &\color{purple}{\left(\frac2\pi\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\sqrt{k}K(k)}{(1-k^2)^{1/4}}\text{d}k =2^{2s-1}\pi^{-s}\Gamma(s)[L_8(s)L_{-8}(s-2)-L_{-8}(s)L_{8}(s-2)]}. \end{aligned}

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Various Theta Products And Relative Lattice Sums

They are listed here. I use some shorten notations:$\vartheta_2=\vartheta_2(q)$ and the like. $\vartheta_n(q^m)(m\ne1)$ won't be omitted. The source function $f(x)$ and corresponding lattice sum are related by $\int_{0}^{\infty}x^{s-1}f(x)\text{d}x =\frac{\Gamma(s)}{\pi^s} L_f(s)$. The third column should be regarded as integrals after direct substitution $x=K^\prime/K$(they may not convergent for specific $s$). So regularization is somewhat significant. We also have $\text{Integral}=\int_{0}^{\infty}x^{s-1}f(x)\text{d}x$. $$\begin{array}{|l|l|l|}\hline \vartheta_2^4& 16\lambda(s)\lambda(s-1) &\left(\frac{2}{\pi}\right) \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{k}{1-k^2}\text{d}k \\ \hline \vartheta_3^4& 8(1+2^{1-s})\zeta(s)\lambda(s-1) &\left(\frac{2}{\pi}\right) \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k(1-k^2)}\text{d}k\\ \hline \vartheta_4^4& 8(2^{2-s}-1)\zeta(s)\lambda(s-1) &\left(\frac{2}{\pi}\right) \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k}\text{d}k \\ \hline \vartheta_2^2\vartheta_3^2& 2^{s+2}\lambda(s)\lambda(s-1) &\left(\frac{2}{\pi}\right) \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{1-k^2}\text{d}k\\ \hline \vartheta_2\vartheta_4^3& 2^{2s}[L_{-8}(s)L_{-8}(s-1)+L_8(s)L_8(s-1)] &\left(\frac{2}{\pi}\right) \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{\sqrt{k}(1-k^2)^{1/4}}\text{d}k \\ \hline \vartheta_2^2\vartheta_4^2& 2^{s+2}\beta(s)\beta(s-1) &\left(\frac{2}{\pi}\right) \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{\sqrt{1-k^2}}\text{d}k \\ \hline \vartheta_2^3\vartheta_4& 2^{2s}[L_{-8}(s)L_{-8}(s-1)-L_8(s)L_8(s-1)] &\left(\frac{2}{\pi}\right) \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{\sqrt{k}}{(1-k^2)^{3/4}}\text{d}k \\ \hline \vartheta_2\vartheta_2(q^2)\vartheta_4^2 &2^{2s}[\beta(s)L_{-8}(s-1)-\lambda(s)L_8(s-1)] &\left ( \frac{2}{\pi} \right ) \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\sqrt{(1-k^\prime)/2}}{\sqrt{k(1-k^2)} }\text{d}k\\ \hline \vartheta_2\vartheta_3(q^2)\vartheta_4^2 &2^{2s}[\beta(s)L_{-8}(s-1)+\lambda(s)L_8(s-1)] &\left ( \frac{2}{\pi} \right ) \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\sqrt{(1+k^\prime)/2}}{\sqrt{k(1-k^2)} }\text{d}k\\ \hline \vartheta_2(q^2)\vartheta_3^2\vartheta_4& 2^{s+1}\beta(s)L_{-8}(s-1) &\left(\frac{2}{\pi}\right) \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{\sqrt{(1-k^\prime)/2}}{k(1-k^2)^{3/4}}\text{d}k \\ \hline \vartheta_2(q^2)\vartheta_3\vartheta_4^2& 2^{s+1}\lambda(s)L_{8}(s-1) &\left(\frac{2}{\pi}\right) \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{\sqrt{(1-k^\prime)/2}}{k\sqrt{1-k^2}}\text{d}k \\ \hline \vartheta_3^2\vartheta_4^2& -2^{3-s}(1-2^{2-s})\zeta(s)\lambda(s-1) &\left(\frac{2}{\pi}\right) \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k\sqrt{1-k^2}}\text{d}k \\ \hline \vartheta_2^6& 2^{s+2}[\beta(s)\lambda(s-2)-\lambda(s)\beta(s-2)] &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{k^2}{1-k^2}K\text{d}k \\ \hline \vartheta_3^6& 16\beta(s)\zeta(s-2)-4\zeta(s)\beta(s-2) &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k(1-k^2)}K\text{d}k\\ \hline \vartheta_4^6& 4\eta(s)\beta(s-2)-16\beta(s)\eta(s-2) &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{\sqrt{1-k^2}}{k}K\text{d}k\\ \hline \vartheta_2^2\vartheta_3^4& 2^{s+2}\beta(s)\lambda(s-2) &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{1-k^2}K\text{d}k \\ \hline \vartheta_2^3\vartheta_3^3& 2^{2s-1}[\beta(s)\lambda(s-2)-\lambda(s)\beta(s-2)] &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{\sqrt{k}}{1-k^2}K\text{d}k \\ \hline \vartheta_2^4\vartheta_3^2& 16\beta(s)\zeta(s-2) &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{k}{1-k^2}K\text{d}k \\ \hline \vartheta_2\vartheta_4(\vartheta_3^4-2\vartheta_2^4)& 2^{2s}[L_8(s)L_{-8}(s-2)+L_{-8}(s)L_8(s-2)] &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1-2k^2}{\sqrt{k}(1-k^2)^{3/4}}K\text{d}k \\ \hline \vartheta_2^2\vartheta_4^4& 2^{s+2}\lambda(s)\beta(s-2) &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1}K\text{d}k\\ \hline \vartheta_2^3\vartheta_4^3& 2^{2s-1}[L_8(s)L_{-8}(s-2)-L_{-8}(s)L_8(s-2)] &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{\sqrt{k}}{(1-k^2)^{1/4}}K\text{d}k \\ \hline \vartheta_2^4\vartheta_4^2& 16\beta(s)\eta(s-2) &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{k}{\sqrt{1-k^2}}K\text{d}k \\ \hline \vartheta_3^2\vartheta_4^4& -4\zeta(s)\beta(s-2) &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k}K\text{d}k \\ \hline \vartheta_3^3\vartheta_4^3& 2^{2-s}[\eta(s)\beta(s-2)-4\beta(s)\eta(s-2)] &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k(1-k^2)^{1/4}}K\text{d}k\\ \hline \vartheta_3^4\vartheta_4^2& 4\eta(s)\beta(s-2) &\left(\frac{2}{\pi}\right)^2\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k\sqrt{1-k^2}}K\text{d}k \\ \hline \vartheta_2^8& 2^{8-s}\lambda(s)\zeta(s-3) & \left(\frac{2}{\pi}\right)^3\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{k^3}{1-k^2}K^2\text{d}k\\ \hline \vartheta_3^8& 32\lambda(s)\lambda(s-3)-16\zeta(s)\eta(s-3) &\left(\frac{2}{\pi}\right)^3\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k(1-k^2)}K^2\text{d}k \\ \hline \vartheta_4^8& -16\zeta(s)\eta(s-3) &\left(\frac{2}{\pi}\right)^3\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1-k^2}{k}K^2\text{d}k \\ \hline \vartheta_2^2\vartheta_3^2(\vartheta_2^4+\vartheta_3^4)& 2^{s+2}\lambda(s)\lambda(s-3) &\left(\frac{2}{\pi}\right)^3\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1+k^2}{1-k^2}K^2\text{d}k \\ \hline \vartheta_2^4\vartheta_3^4& 16\lambda(s)\zeta(s-3) &\left(\frac{2}{\pi}\right)^3\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{k}{1-k^2}K^2\text{d}k \\ \hline \vartheta_2^4\vartheta_4^4& 16\lambda(s)\eta(s-3) &\left(\frac{2}{\pi}\right)^3\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} kK^2\text{d}k \\ \hline \vartheta_3^4\vartheta_4^4& -2^{4-s}\zeta(s)\eta(s-3) &\left(\frac{2}{\pi}\right)^3\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k}K^2\text{d}k \\ \hline \vartheta_2^{10} & \frac{2^{s+2}}5\beta(s)\lambda(s-4)+\frac{2^{s+2}}5\lambda(s)\beta(s-4)-\frac{2^{6-s}}{5}L_{1/2,1}(s) &\left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{k^4 }{1-k^2}K^3\text{d}k \\ \hline \vartheta_3^{10}& \frac45\zeta(s)\beta(s-4)+\frac{64}5\beta(s)\zeta(s-4)+\frac25L_{1,1}(s) &\left(\frac{2}{\pi}\right)^4\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k(1-k^2)}K^3\text{d}k \\ \hline \vartheta_4^{10} & -\frac45\eta(s)\beta(s-4)-\frac{64}5\beta(s)\eta(s-4)-\frac{2^{s+1}}{5}L_{1,-1}(s) &\left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\left ( 1-k^2 \right )^{3/2} }{k}K^3\text{d}k \\ \hline \vartheta_2\vartheta_3\vartheta_4^8 &16L_{1/2,1}(s):=16{\sum_{(m,n)\in\mathbb{Z}^2}^{}}\frac{(m+1/2-ni)^4}{ [(m+1/2)^2+n^2]^s} & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1-k^2}{\sqrt{k}}K^3\text{d}k\\ \hline \vartheta_2^2\vartheta_3^4\vartheta_4^4 &2^{5-s}L_{1/2,1}(s) & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1}K^3\text{d}k\\ \hline \vartheta_2^2\vartheta_3^8 &\frac{2^{s+2}}5\beta(s)\lambda(s-4)+\frac{2^{7-s}}5L_{1/2,1}(s) & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1}\frac{1}{1-k^2}K^3\text{d}k\\ \hline \vartheta_2^2\vartheta_4^8 &\frac{2^{s+2}}5\lambda(s)\beta(s-4)+\frac{2^{7-s}}5L_{1/2,1}(s) & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1}(1-k^2)K^3\text{d}k\\ \hline \vartheta_2^4\vartheta_4^6 &\frac{64}{5}\beta(s)\eta(s-4)+\frac{2^{s}}{5}L_{1,-1}(s) & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} k\sqrt{1-k^2} K^3\text{d}k\\ \hline \vartheta_2^4\vartheta_3^2\vartheta_4^4& L_{1,1}(s):={\sum_{(m,n)\in\mathbb{Z}^2}}^\prime\frac{(m-ni)^4}{(m^2+n^2)^s} &\left(\frac{2}{\pi}\right)^4\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1}kK^3\text{d}k \\ \hline \vartheta_2^4\vartheta_3^4\vartheta_4^2 & 2^sL_{1,-1}(s) & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{k}{\sqrt{1-k^2}}K^3\text{d}k\\ \hline \vartheta_2^{5}\vartheta_3^5 & \frac{2^{2s-3}}5\beta(s)\lambda(s-4)+\frac{2^{2s-3}}5\lambda(s)\beta(s-4)-\frac{2}{5}L_{1/2,1}(s) &\left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{k^{3/2} }{1-k^2}K^3\text{d}k\\ \hline \vartheta_2^6\vartheta_3^4 &\frac{2^{s+2}}5\beta(s)\lambda(s-4)-\frac{2^{5-s}}5L_{1/2,1}(s) &\left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{k^2}{1-k^2}K^3\text{d}k\\ \hline \vartheta_2^6\vartheta_4^4 &-\frac{2^{s+2}}5\lambda(s)\beta(s-4)+\frac{2^{5-s}}{5}L_{1/2,1}(s) & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} k^2K^3\text{d}k\\ \hline \vartheta_2^8\vartheta_3^2 &\frac{64}5\beta(s)\zeta(s-4)-\frac{4}5L_{1,1}(s) &\left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{k^3}{1-k^2}K^3\text{d}k\\ \hline \vartheta_2^8\vartheta_4^2 &-\frac{64}{5}\beta(s)\eta(s-4)+\frac{2^{s+2}}{5}L_{1,-1}(s) & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{k^3}{\sqrt{1-k^2} }K^3\text{d}k\\ \hline \vartheta_2^8\vartheta_3\vartheta_4 &16L_{1,-1}(s):=16{\sum_{(m,n)\in\mathbb{Z}^2}^{}}^\prime(-1)^n\frac{(m-ni)^4}{ (m^2+n^2)^s} & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{k^3}{\left ( 1-k^2 \right )^{3/4} }K^3\text{d}k\\ \hline \vartheta_3^2\vartheta_4^8 &\frac45\zeta(s)\beta(s-4)-\frac{4}5L_{1,1}(s) & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1-k^2}{k}K^3\text{d}k\\ \hline \vartheta_3^4\vartheta_4^6 &-\frac45\eta(s)\beta(s-4)-\frac{2^{s}}5L_{1,-1}(s) & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\sqrt{1-k^2} }{k}K^3\text{d}k\\ \hline \vartheta_3^5\vartheta_4^5 & -\frac{2^{2-s}}5\eta(s)\beta(s-4)-\frac{2^{6-s}}5\beta(s)\eta(s-4)-\frac{2}{5}L_{1,-1}(s) &\left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\left ( 1-k^2 \right )^{1/4} }{k}K^3\text{d}k\\ \hline \vartheta_3^6\vartheta_4^4 &\frac45\zeta(s)\beta(s-4)+\frac{1}5L_{1,1}(s) & \left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{k}K^3\text{d}k\\ \hline \vartheta_3^8\vartheta_4^2 &-\frac45\eta(s)\beta(s-4)+\frac{2^{s+2}}5L_{1,-1}(s) &\left ( \frac{2}{\pi} \right )^4 \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{k\sqrt{1-k^2}}K^3\text{d}k\\ \hline \end{array}$$

Answer 2

$$\begin{array}{|l|l|l|}\hline \textbf{Source}&\textbf{Lattice Sum}&\textbf{Integral}\\ \hline \vartheta_2& 2^{2s+1}\lambda(2s)&\sqrt{\frac{\pi}{2} } \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{\sqrt{k}(1-k^2)K^{3/2} }\text{d}k\\ \hline \vartheta_3&2\zeta(2s)&\sqrt{\frac{\pi}{2} } \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{k(1-k^2)K^{3/2} }\text{d}k\\ \hline \vartheta_4& -2\eta(2s) &\sqrt{\frac{\pi}{2} } \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{k(1-k^2)^{3/4}K^{3/2} }\text{d}k \\ \hline (\vartheta_2\vartheta_4)^{1/2}& 2^{3s+1/2}L_8(2s)&\sqrt{\frac{\pi}{2} } \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{k^{3/4}(1-k^2)^{7/8}K^{3/2} }\text{d}k\\ \hline (\vartheta_2\vartheta_3\vartheta_4)^{1/3}&2^{1/3}12^sL_{12}(2s) & \sqrt{\frac{\pi}{2} } \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{k^{5/6}(1-k^2)^{11/12}K^{3/2} }\text{d}k \\ \hline \vartheta_3^{1/2}(\vartheta_2\vartheta_3\vartheta_4)^{1/6}& 2^{1/6}24^s L_{24}(2s)&\sqrt{\frac{\pi}{2} } \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{k^{11/12}(1-k^2)^{23/24}K^{3/2} }\text{d}k \\ \hline \vartheta_2^2& 2^{s+2}\lambda(s)\beta(s) &\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1}\frac{1}{(1-k^2)K }\text{d}k \\ \hline \vartheta_3^2& 4\zeta(s)\beta(s) &\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{1}{k(1-k^2)K }\text{d}k \\ \hline \vartheta_4^2& -4\eta(s)\beta(s)&\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1}\frac{1}{k\sqrt{1-k^2}K}\text{d}k\\ \hline \vartheta_2\vartheta_3& 2^{2s+1}\lambda(s)\beta(s)& \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1}\frac{1}{\sqrt{k}(1-k^2)K }\text{d}k \\ \hline \vartheta_2\vartheta_4& 2^{2s+1}L_8(s)L_{-8}(s) &\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1}\frac{1}{\sqrt{k}(1-k^2)^{3/4}K }\text{d}k \\ \hline \vartheta_2\vartheta_2(q^2)& 2^{2s}(\lambda(s)L_{-8}(s)-\beta(s)L_8(s)) &\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\sqrt{(1-k^\prime)/2}}{\sqrt{k}(1-k^2)K }\text{d}k\\ \hline \vartheta_2\vartheta_2(q^3)& 4\lambda(s) L_{-12}(s) &- \\ \hline \vartheta_2\vartheta_3(q^2)& 2^{2s}(\lambda(s)L_{-8}(s)+\beta(s)L_8(s)) &\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\sqrt{(1+k^\prime)/2}}{\sqrt{k}(1-k^2)K }\text{d}k\\ \hline \vartheta_2(q^2)\vartheta_3& 2^{s+1}\lambda(s)L_{-8}(s) &\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{\sqrt{(1-k^\prime)/2}}{k(1-k^2)K }\text{d}k \\ \hline \vartheta_2(q^2)\vartheta_4& 2^{s+1}\beta(s)L_8(s) &\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{\sqrt{(1-k^\prime)/2}}{k(1-k^2)^{3/4}K }\text{d}k \\ \hline \vartheta_3\vartheta_3(q^2)& 2\zeta(s)L_{-8}(s) &\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{\sqrt{(1+k^\prime)/2}}{k(1-k^2)K }\text{d}k\\ \hline \vartheta_3\vartheta_3(q^3)& 2(1+2^{1-2s})\zeta(s)L_{-3}(s) &-\\ \hline \vartheta_3\vartheta_4& -2^{2-s}\eta(s)\beta(s)&\int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1}\frac{1}{k(1-k^2)^{3/4}K}\text{d}k\\ \hline \vartheta_3(q^2)\vartheta_4& -2\eta(s)L_{-8}(s)& \int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} \frac{\sqrt{(1+k^\prime)/2}}{k(1-k^2)^{3/4}K }\text{d}k\\ \hline \vartheta_4\vartheta_4(q^3)& 2(1+2^{1-2s})\zeta(s)L_{-3}(s)- 4\lambda(s) L_{-12}(s) &- \\ \hline \vartheta_2\vartheta_3\vartheta_4& 2^{2s+1}\beta(2s-1) &\sqrt{\frac{2}{\pi} } \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{\sqrt{k}(1-k^2)^{3/4}\sqrt{K}}\text{d}k \\ \hline \vartheta_2\vartheta_3(\vartheta_2\vartheta_3\vartheta_4)^{1/3}& 3^s(1+2^{2-2s})L_{-3}(2s-1) & \sqrt{\frac{2}{\pi} } \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k^{1/3}(1-k^2)^{11/12}\sqrt{K}}\text{d}k\\ \hline \vartheta_2\vartheta_4(\vartheta_2\vartheta_3\vartheta_4)^{1/3}& 3^sL_{-3}(2s-1) &\sqrt{\frac{2}{\pi} } \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k^{1/3}(1-k^2)^{2/3}\sqrt{K}}\text{d}k\\ \hline \vartheta_3^2(\vartheta_2\vartheta_4)^{1/2}& 8^sL_{-8}(2s-1) &\sqrt{\frac{2}{\pi} } \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k^{3/4}(1-k^2)^{7/8}\sqrt{K}}\text{d}k \\ \hline \vartheta_3^{5/2}(\vartheta_2\vartheta_3\vartheta_4)^{1/6}& 24^sL_{-24}(2s-1) &\sqrt{\frac{2}{\pi} } \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k^{11/12}(1-k^2)^{23/24}\sqrt{K}}\text{d}k \\ \hline \vartheta_4^{5/2}(\vartheta_2\vartheta_3\vartheta_4)^{1/6}& 24^s(1+2^{1-2s})L_{-3}(2s-1) &\sqrt{\frac{2}{\pi} } \int_{0}^{1} \left ( \frac{K^\prime}{K}\right )^{s-1} \frac{1}{k^{11/12}(1-k^2)^{2/3}\sqrt{K}}\text{d}k\\ \hline \end{array}$$