Summing up all digamma roots

What an intriguing problem! It required a bit of detective work as I'm not exactly an expert in this field.

There is a useful formula for the digamma function, $\frac{\psi'(z)}{\psi(z)}-\psi(z)=2\gamma-z\sum_{k=0}^{\infty}\frac{1}{a_k^2-a_kz}$ where the $a_k$ are the zeros of $\psi(z)$ . Evaluating this at $z=2$ and observing that $\psi(2)=1-\gamma$ and $\psi'(z)=\sum_{n=0}^{\infty}\frac{1}{(z+n)^2}$ so $\psi'(2)=\frac{\pi^2}{6}-1$ , we find that the sum we seek comes out to be $\frac{1-\frac{\gamma^2}{2}-\frac{\pi^2}{12}}{1-\gamma}$

so that the answer is $\boxed{18}$