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

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