On zeros of Eisenstein series for genus zero Fuchsian groups

Let Gamma <= SL2(R) be a genus zero Fuchsian group of the first kind with infinity as a cusp, and let E-2k(Gamma) be the holomorphic Eisenstein series of weight 2k on Gamma that is nonvanishing at infinity and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on Gamma, and on a choice of a fundamental domain F, we prove that all but possibly c(Gamma,F) of the nontrivial zeros of E-2k(Gamma) lie on a certain subset of {z epsilon h : j(Gamma)(z) is an element of R}. Here c(Gamma, F) is a constant that does not depend on the weight, h is the upper half-plane, and j(Gamma) is the canonical hauptmodul for Gamma.