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

Coefficients of the monodromy matrices of one-parameter families of double octic Calabi-Yau threefolds at a half-conifold point