Function $f:\mathbb{C} \to \mathbb{C}$, defined as
$$f(z) = x^{-z} (\sin(\pi z))^{-2} \left [ \left (-\frac{z}{2} \right )! \right ]^{-4} \left [\left (\frac{z}{2} - \frac12 \right)! \right ]^{-2}$$
where $x$ is a positive constant, $z\in \mathbb{C}$, and where $z!$ is the $\Gamma(z+1)$,
has double-poles at $z = -1, -3, -5 \ldots$ and at $z = 2, 4, 6 \ldots$
Can you please help me compute the residues of those poles?