Let F/Q be a totally real number field of degree n. We explicitly evaluate a certain sum of rational functions over a infinite fan of F-rational polyhedral cones in terms of the norm map N: F ¨ Q. This completes Sczech fs combinatorial proof of Satake fs conjecture connecting the special values of L-series associated to cusp singularities with intersection numbers of divisors in their toroidal resolutions [Sc2].