Variants of the Basel problems The Basel problem asserts $\sum\frac{1}{n^2}=\frac{\pi^2}{6}$, the well-known proof uses the Fourier series of $f(x)=x^2$.
Question: has somebody computed sums of the form $$\sum\frac{1}{(n+\frac{1}{2})^2}$$
or
$$\sum\frac{1}{n^2+1}?$$
The approach via Fourier series seems not to generalize.

Yes actually we know the exact value of $$S = \sum_{m=1}^{+\infty} \frac{1}{am^2+b},$$ with $a,b >0$. This is computed using the classical residue method with $\pi \operatorname{cotg}(\pi z).$ You will eventually get $$S = \frac{1}{2} \left( \frac{\pi \coth(\sqrt{\frac{b}{a}}\pi)}{\sqrt{ab}}-\frac{1}{b}\right).$$
