The O(4) symmetry of the nonrelativistic hydrogen atom is exploited in conjunction with Green's-function methods as a tool for the solution of Coulomb problems. The Bethe logarithm, the principal contribution to the Lamb shift, is treated by this means, and a rapidly convergent series is obtained. This is used to evaluate the Bethe logarithm of the 1S, 2S, and 2P states, and a discrepancy with the most precise previously published value for the 1S state is indicated. The new value is 2.9841285 with an estimated uncertainty of +/-3 in the last figure. The discrepancy is beyond the reach of experimental observation at present.