In mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields.
The local L-factor at a finite place v is similarly given by the characteristic polynomial of a Frobenius element at v acting on the v-inertial invariants of the v-adic realization of the motive.
For infinite places, Jean-Pierre Serre gave a recipe in (Serre 1970) for the so-called Gamma factors in terms of the Hodge realization of the motive.
It is also known (Scholl 1990), for example, that a motive can be attached to a newform (i.e. a primitive cusp form), hence their L-functions are motivic.
Several conjectures exist concerning motivic L-functions.