meijer g function explicit form

Question

Can the following case of the Meijer G-function $$ G_{2,3}^{3,1}\left(z\left|\begin{smallmatrix}0,1\\ 0,0,0\end{smallmatrix}\right.\right) $$ be expressed more explicitly (in terms of other special functions, but involving no infinite integrals)? This question seems to be related to this one, this one, and this one.

Answer 1

With some help from Mathematica: $$G_{2,3}^{3,1}\left(z\left| \begin{smallmatrix} 0,1 \\ 0,0,0 \\ \end{smallmatrix} \right.\right) = -2 z \, {_3F_3}\left(\begin{smallmatrix}1,1,1\\2,2,2\end{smallmatrix}\middle|\,z\right)-\frac{1}{2} \frac{\partial^2}{\partial t^2}\left.{_1F_1}\left(\begin{smallmatrix}t\\1\end{smallmatrix}\middle|\,z\right)\right|_{t=0}+ \left(\operatorname{Ei}(z)+\tfrac{1}{2}\ln\left(\tfrac1z\right)\right)\left(\gamma+\ln(z)\right)-\frac{\gamma}{2}\ln (z)-\frac{\gamma^2}{2}+\frac{\pi ^2}{4}.$$