Mock period functions and non-critical values of L-functions

We discuss how non-critical values of L-functions can be
incorporated into a cohomological setting analogous to the Eichler cocycle
setting associated to critical values by Manin. The "cocycle" attached to
non-critical values has a structure resembling that of the non-holomorphic
part of a harmonic weak Maass form. We formalize this similarity by
interpreting the "cocycle" in the context of mock modular forms.