In a physics problem that I assigned in one of my classes recently, the following series arises: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})} $$ From the physical context, one can argue that the series should converge to $$ S = - \frac{\pi^2}{96}, $$ and numerically calculating the first 10–20 terms of this series seems to show rapid convergence to this value.
Is it possible to prove this statement without appealing to a physical argument? I honestly have no idea how to begin tackling a problem like this.
(For the curious: the physical problem where this arises is the solution via separation of variables of Laplace's equation inside a cube, where the function goes to zero on five of the faces and a non-zero value $f_0$ on the sixth. This series arises from looking at the value of the function at the center point, and an argument via superposition of solutions then shows that the value at the center should be $f_0/6$.)