A correspondence theorem for L-functions and partial differential operators

Given an L-function F(s) from the extended Selberg class, we associate a function Phi(F)(x, y). We show that the functions Phi(F)(x, y) are, in the general case, the analogs of the modular forms associated with the GL(2) L-functions. Indeed, we prove that every Phi(F)(x, y) is eigenfunction of a certain partial differential operator. Moreover, we prove a general correspondence theorem for such L-functions involving the functions Phi(F)(x, y).