Barnett and Coulson's zeta-function method (M. P. Barnett and C. A. Coulson, Philos. Trans. R. Soc., Lond. A 1951, 243, 221) is one of the main sources of algorithms for the solution of multicenter integrals with Slater-type orbitals. This method is extended here from single functions to two-center charge distributions, which are expanded at a third center in terms of spherical harmonics times analytical radial factors. For s-s distributions, the radial factors are given by a series of factors corresponding to the translation of s-type orbitals. For distributions with higher quantum numbers, they are obtained from those of the s-s distributions by recurrence. After analyzing the convergence of the series, a computational algorithm is proposed and its practical efficiency is tested in three-center (AB/CC) repulsion integrals. In cases of large basis sets, the procedure yields about 12 correct significant figures with a computational cost of a few microseconds per integral.