We have $$\frac {\Gamma (a)}{2^a}=\int _{(c)}\Gamma (s)\Gamma (a-s)ds,$$ see e.g. Exercise C.23 of Montgomery and Vaughan's "Multiplicative Number Theory".

I would like a similar formula for $\int _{(c)}\Gamma (s)\Gamma (1-s)\Gamma (a+2-s)\Gamma (s-a)ds$.

I've looked at a few reference books for Mellin transforms but can't find this one. It seems to be a particular case of the Meijer $G$-function (https://en.wikipedia.org/wiki/Meijer_G-function), with arguments $G_{2,2}^{2,2}\left (\begin {array}{ll}1,&a+1\\ 1,&a+2\end {array}\right )$ however when I put this in WolframAlpha I don't get anything out.

I suspect that the integral is something like $p(a)/\sin (\pi a)$ with $p$ a quadratic polynomial, but it seems I'm not proficient enough in Mellin transforms to be sure I'm not misunderstanding or miscalculating something.

We have $$\frac {\Gamma (a)}{2^a}=\int _{(c)}\Gamma (s)\Gamma (a-s)\,ds,$$ see e.g. Exercise C.23 of Montgomery and Vaughan's "Multiplicative Number Theory".

I would like a similar formula for $$\int_{(c)} \Gamma (s)\Gamma (1-s)\Gamma (a+2-s)\Gamma (s-a)\,ds.$$

I've looked at a few reference books for Mellin transforms but can't find this one. It seems to be a particular case of the Meijer $G$-function (https://en.wikipedia.org/wiki/Meijer_G-function), with arguments $G_{2,2}^{2,2}\left (\begin {array}{ll}1,&a+1\\ 1,&a+2\end {array}\right )$ however when I put this in WolframAlpha I don't get anything out.

I suspect that the integral is something like $p(a)/\sin (\pi a)$ with $p$ a quadratic polynomial, but it seems I'm not proficient enough in Mellin transforms to be sure I'm not misunderstanding or miscalculating something.