Interpolated Apéry numbers, quasiperiods of modular forms, and motivic gamma functions

We extend the definitions of the sequences used by Apery in his proof of the irrationality of (3) to non-integral values of the index and relate the value with index - 1=2 to the central value of the L-series of the unique normalized cusp form of weight 4 on Γ_0(8). We also discuss the notion of quasiperiods of modular forms and relate the Apery numbers of other halfintegral indices to these. We further explain the conjectural relationship of the Taylor expansion around 0 of a dierent interpolation of the Apery numbers to a generalized version of the Gamma Conjecture, and discuss interpretations of the various results with families of Calabi-Yau manifolds, mirror symmetry, and motivic gamma functions