Help with an integral involving zeta and digamma

Question

Consider the integral: $$f(s)=\frac{1}{2\pi i}\int_{\tau-i\infty}^{\tau+i\infty}\frac{\zeta(z)}{z}\left[\psi\left(\frac{z}{s} \right ) +\frac{s}{2z}-\log\left(\frac{z}{s} \right )\right ]dz\;\;\;\;(\tau<0)$$ $\zeta(\cdot)$ being the Riemann zeta function, and $\psi\left(\cdot\right)$ being the Digamma function. We wish for an explicit evaluation of integral, possibly with a proof that $f(s)$ has an analytic continuation to the whole complex plane, except for a branch cut along the $s$ imaginary axis.