Is it possible to obtain any closed-form expression for the infinite sum $$\sum_{n=1}^{\infty}\frac{\log(n)}{n^{2}}$$ by Residue calculus?
My thought was to try to integrate $$f(z) =\frac{\pi\log(z)}{z^{2}\tan(\pi z)}$$ over the rectangle $R_{M}$ with vertices $(M+\frac{1}{2})(\pm 1 \pm i)$