I want to calculate residue at the poles for $\frac{\zeta(s)}{\zeta(2s)}.$ For pole of numerator $s=1$ I have calculated the residue. I am having trouble at the zeros of denominator. Basically I am interested in this limit $$ Res_{s= \frac{1}{2}\rho}\frac{\zeta(s)}{\zeta(2s)} = \lim_{s\to \frac{1}{2}\rho } \frac{(s- \frac{1}{2}\rho )\zeta(s)}{\zeta(2s)} $$ where $\rho$ are the non-trivial zeros of $\zeta(s).$
Any help will be appreciated. Thanks.