You can find a huge collection of examples in this website:
As for a concrete example, the critical temperature of the Bose-Einstein condensate is
$$T=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi \hbar^2}{mk_B}$$
where $\zeta(s)$ is the Riemann zeta function.
Particularly after the work of Ketterle, this seems as "real" and non-speculative as it gets.