So I need to calculate the following integral (which I know would somehow involve a Gamma function, except I'm not sure how):
$$ \idotsint \prod d^3 p_i \exp(-\beta A \vert p_i \vert^s) $$
where s > 0, and each $p_i$ is independent of the others, and i runs from 0 to N.
(to those interested, this comes from trying to calculate the partition function of some general hamiltonian using the canonical ensemble (statistical mechanics))
Each $i$-integral can be independently evaluated: $$ \mbox{as}\ {4\left(A\beta\right)^{-3/s}\,\pi\,\Gamma\left(3/s\right) \over s}. $$ $$ \mbox{So, the final result must be}\ \bbox[15px, border:1px solid navy]{\color{#44f}{\left[{4\left(A\beta\right)^{-3/s}\,\pi\,\Gamma\left(3/s\right) \over s}\right]^{N}}} $$