  Doran and Morgan have introduced certain rational basis for the monodromy
group of the Picard-Fuchs operator of a hypergeometric family of Calabi-Yau
threefolds. In this paper we compute numerically the transition matrix between
a generalization of the Doran-Morgan basis and the Frobenius basis at a
half-conifold point of a one-parameter family of double octic Calabi-Yau
threefolds. We identify the entries of this matrix as rational functions in the
special values $L(f,1)$ and $L(f,2)$ of the corresponding modular form $f$ and
one constant. We also present related results concerning the rank of the group
of period integrals generated by the action of the monodromy group on the
conifold period.
