  We consider the semiclassical limit of the vacuum Virasoro block describing
the diagonal 4-point correlation functions on the sphere. At large central
charge c, after exponentiation, it depends on two fixed ratios h_H/c and h_L/c,
where h_{H, L} are the conformal dimensions of the 4-point function operators.
The semiclassical block may be expanded in powers of the light ratio h_L/c and
the leading non-trivial (linear) order is known in closed form as a function of
h_H/c. Recently, this contribution has been matched against AdS_3 gravity
calculations where heavy operators build up a classical geometry corresponding
to a BTZ black hole, while the light operators are described by a geodesic in
this background. Here, we compute for the first time the next-to-leading
quadratic correction O((h_L/c)^{2}), again in closed form for generic heavy
operator ratio h_H/c. The result is a highly non-trivial extension of the
leading order and may be relevant for further refined AdS_{3}/CFT_{2} tests.
Applications to the two-interval R\'enyi entropy are also presented.
