  In the $(2,5)$ minimal model, the partition function for genus $g=2$ Riemann
surfaces is given by a $5$-tuple of functions with appropriate transformation
under the mapping class group. These functions generalise the two
Rogers-Ramanujan functions for the torus. Their expansions around a locus of
surfaces with conical singularities in the interior of the $g=2$ Siegel upper
half plane are obtained in terms of standard modular forms. The dependence on
the metric is controlled by a canonical choice of flat surface metrics. In the
alternative case where a handle of the $g=2$ surface is pinched, our method
requires knowledge of the two-point function of the fundamental lowest-weight
vector in the non-vacuum representation of the Virasoro algebra, for which we
derive a $3$\ts{rd} order ODE. In order to make the paper more accessible to
mathematicians, the exposition includes a short introduction to conformal field
theory on Riemann surfaces, which may be of independent interest.
