This identity is still not proven:
$$\sum_{n=0}^\infty \left(\frac{1}{(7n+1)^2}+\frac{1}{(7n+2)^2}-\frac{1}{(7n+3)^2}+\frac{1}{(7n+4)^2}-\frac{1}{(7n+5)^2}-\frac{1}{(7n+6)^2}\right)=\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2} \log \left| \frac{\tan t + \sqrt{7} }{\tan t - \sqrt{7} } \right| dt$$
It arose from physical applications.
