Summary: We construct $p$-adic L-functions for automorphic representations of $\GL$_2 of a number field $F$ , and show that the corresponding $p$-adic L-function of a modular elliptic curve $E$ over $F$ has an extra zero at the central point for each prime above $p$ at which $E$ has split multiplicative reduction, a part of the exceptional zero conjecture.

11F41, 11F67, 11F70, 11G40

p-adic L-function, automorphic forms, exceptional zero conjecture, Mazur-Tate-teitelbaum conjecture