Working on algorithms related to number theoretical function calculation performance improvement and accidentally discovered the following for Dedekind $\eta$ function: $$-\int_1^{\infty } \left(t^{-s-1}+t^{s-1}\right) \left(\frac{\pi t}{12}+\log (\eta (i t))\right) \, dt=\frac{\Gamma (s) \zeta (s+1) \zeta (s)}{(2 \pi )^s}-\frac{(\pi -3) s^2+3}{6 s^2 \left(s^2-1\right)}$$ which may become interesting tool to investigate the zeros of $\zeta$-function.
I need a help in calculation of the left side integral effectively and understanding of the range of convergence of it.
