I have to prove (if $\gamma \ne 0$) that there is a analytic continuation for $\Re s >0$ of the function $$f(s)=\frac{\zeta (s)^2 \zeta(s-i\gamma )\zeta(s+i\gamma ) }{\zeta(2s)} $$ and that this continuation has possible singularities at $s=1,~s=1\pm i\gamma$ and $\Re s =1/2$.
The analytic continuation is clear since $\zeta $ has such a continuation - is this sufficient? And concerning the singularities: $s=1,~s=1\pm i\gamma$ is obvious since $\zeta$ has a pole at $s=1$. But what about $\Re s =1/2$?