One says the the p-adic L-function Lp (f,s) of a modular form f has a trivial zero if the interpolation property forces Lp (f,s) to vanish at some integer s = m. This phenomenon was first studied at 1980’s by Mazur, Tate and Teitelbaum. For modular forms of even weight they formulated a precise conjecture about the value of the derivative L'p (f,s) at s = m. This conjecture was proved by Greenberg-Stevens (using p-adic families of modular forms) and by Kato-Kurihara-Tsuji (using Euler systems) around 1998. In this talk we formulate and prove an analog of this result for modular forms of odd weight. Our key tool is the theory of ()-modules which allows to define the L-invariant for a large class of p-adic representations.