p-adic L functions are analytical functions of p-adic characters that, one way or another, interpolate special values of classical (complex) L functions. The first such examples were the p-adic L functions of Kubota and Leopoldt [K-Le], interpolating Dirichlet L series. Manin and Vishik [M-V] and Katz [Kl] constructed p-adic L functions which interpolate special values of Hecke L series associated with a quadratic imaginary field K, in which p splits. (To fix notation write p = pp^-). Their work gave p-adic interpolation of the Hasse-Weil zeta function of certain elliptic curves with complex multiplication and good ordinary reduction at p (those whose division points generate abelian extensions of K). The p-adic L function of Manin-Vishik and Katz is the first object studied in this work.