I'm considering the complex integration
$$ \int_{\sigma=-1+\epsilon} |\zeta^2(1+2s)\Gamma(s)|\mathrm{d}s, $$
where $s=\sigma+i\tau$, $\sigma,\tau \in \mathbb R$. Since the zeta function increase polynomially if $\sigma$ is uniformly bounded and the gamma function decrease exponentially by the Stirling's formula, I believe this integration does not diverge i.e. it has a finite value. However, I couldn't prove my claim rigorously.