I want to know if there can be a general statement about the poles (Laurent expansion) of such products of Gamma functions as a function of $p \in \mathbb{R}$ in the limit $\epsilon \rightarrow 0$,

$\Gamma[ \frac{(-1-2p)}{2} - \epsilon ]\Gamma[\frac{-1+p}{2} + \epsilon]\Gamma[\frac{(5+p)}{2} + \epsilon]$

For any given $p$ such that the $p$ dependent part of the argument is a negative integer one can do the usual Laurent expansion of the Gamma function in $\epsilon$ for each of the 3 factors and then multiply. But what can be said about the Laurent expansion in general as a function of $p$?

I wish one could write down the Laurent expansion in $\epsilon$ as a function of $p$!

One sees that there are these special cases like if $p$ is such that there are two integers $N$ and $M$ satisfying, $p= 1-2N^2 = -5 -2M^2$ then the later two Gamma function can have poles simultaneously. (like $N=2, M =1, p= -7$) Existence of such special $p$ naively seems to make things more tricky.