Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$?

Question

Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$?

where res means the residue of the function?

I know $\zeta(s)$ has a pole at $s=1$ but i can't see where the factor of a $\frac{1}{2}$ comes from in the answer

Answer 1

$\frac{1}{2s-1}=\frac{1}{2}\frac{1}{s-\frac{1}{2}}$ as the residue is computed at $\frac{1}{2}$ not at $1$