We introduce an analog of part of the Langlands-Shahidi method to p-adic SL(2), constructing reciprocals of certain p-adic L-functions using the non constant terms of the Fourier expansions of Eisenstein series. Using p regular, we give explicit p-adic measures whose Mellin transforms are reciprocals of Dirichlet L- functions (plus a chosen multiple of Haar measure necessary for boundedness). This is joint work with S. Miller, A. Panchishkin, and F. Shahidi.