  Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for
$\zeta(3)$ 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.
