  We introduce a new technique of completion for 1-cohomology which parallels
the corresponding technique in the theory of mock modular forms. This technique
is applied in the context of non-critical values of L-functions of GL(2,Q) cusp
forms. We prove that a generating series of non-critical values can be
interpreted as a mock period function we define in analogy with period
polynomials. Further, we prove that non-critical values can be encoded into a
sesquiharmonic Maass form. Finally, we formulate and prove an
Eichler-Shimura-type isomorphism for the space of mock period functions.
