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We introduce a new technique of completion for 1-cohomology which parallelsthe corresponding technique in the theory of mock modular forms. This techniqueis applied in the context of non-critical values of L-functions of GL(2,Q) cuspforms. We prove that a generating series of non-critical values can beinterpreted as a mock period function we define in analogy with periodpolynomials. Further, we prove that non-critical values can be encoded into asesquiharmonic Maass form. Finally, we formulate and prove anEichler-Shimura-type isomorphism for the space of mock period functions.

Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions