We know the Euler product.
$$\zeta (s)=\prod_{p}\frac{p^{s}}{p^{s}-1}$$
I wonder if there is formula or any kind of work for this kind of prime product below?
$$\prod_{p\equiv a \ (mod \ b)}\frac{p^{s}}{p^{s}-1}$$
Where $a$ and $b$ are coprime.
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2For $(a,b)=(1,2)$ the answer should be obvious. For $(a,b)=(1,4)$ and $(3,4)$ see OEIS A$88539$ and OEIS A$243379$. See also this related question. – Lucian Jan 09 '16 at 15:12
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2and see the Dirichlet characters and L-function which are $$\prod_p \frac{1}{\displaystyle 1- p^{-s} e^{\textstyle \frac{2 i \pi}{b} \ln_{a,b}(p)}}$$ where $ \ln_{a,b}(p)$ is the discrete logarithm modulo $b$ – reuns Jan 09 '16 at 20:54