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

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

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