We introduce and study the overconvergent cohomology adapted to the
Shalika setting. Then we describe how to evaluate this cohomology in order to produce distributions over the expected Galois group. Moreover, we verify that this overconvergent evaluation interpolates the classical evaluations explained in the first lecture. Another consequence of this method is the control of the growth of the distribution obtained. The $p$-adic $L$-functions are, as usual, the Mellin transform of these distributions.