<a href="http://www.emis.ams.org/journals/EM/expmath/volumes/12/12.4/Guillera.pdf">Gourevitch's conjecture:</a>
$$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7 = \frac{32}{\pi^3}.$$