Interpolated Sequences and Critical L-Values of Modular Forms

Recently, Zagier expressed an interpolated version of the Apéry numbers for 𝜁(3) in terms of a critical L-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier’s six sporadic sequences are essentially critical L-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown’s cellular integrals and critical L-values of modular forms of odd weight.