For every triple F , K, p where F is a classical elliptic eigenform, K is a quadratic imaginary field and p is an odd prime integer which is not split in K, I will explain how to attach a p-adic L-function which interpolates the algebraic parts of the special values of the complex L-functions of F twisted by algebraic Hecke characters of K such that the p-part of their conductor is p^n (for n large enough and explicitly determined). This construction extends a classical construction of N. Katz for F an Eisenstein series and of Bertolini–Darmon–Prasanna for F a cuspform when p is split in K.
This is joint work with Adrian Iovita.