L-functions of $S_3(\G_2(2,4,8))$

The space of Siegel cuspforms of degree $2$ of weight $3$ with respect to the congruence subgroup $\G_2(2,4,8)$ was studied by van Geemen and van Straten in Math. computation. {\bf 61} (1993). They showed the space is generated by six-tuple products of Igusa $\th$-constants, and all of them are Hecke eigenforms. They gave conjecture on the explicit description of the Andrianov $L$-functions. In J. Number Theory. {\bf 125} (2007), we proved some conjectures by showing that some products are obtained by the Yoshida lift, a construction of Siegel eigenforms. But, other products are not obtained by the Yoshida lift, and our technique did not work. In this paper, we give proof for such products. As a consequence, we determine automorphic representations of O(6), and give Hermitian modular forms of SU(2,2) of weight $4$. Further, we give non-holomorphic differential threeforms on the Siegel threefold with respect to $\G_2(2,4,8)$.