  We study (compactly supported) cohomology of character stacks of punctured
Riemann surfaces with prescribed semisimple local monodromies at punctures. In
the case of generic local monodromies, the cohomology of these character stacks
has already been studied by Hausel, Letellier and Rodriguez-Villegas and by
Mellit. In this paper, we extend the results of Hausel, Letellier and
Rodriguez-Villegas to the non-generic case. In particular, we compute the
E-series and we give a conjectural formula for the mixed Poincar\'e series.
Moreover, we verify our conjecture in the case of the projective line with 4
punctures and a certain choice of a non-generic quadruple.
