L-functions of S3(Γ(4,8))

We prove most of B. van Geemen and D. van Straten's conjectures on the explicit description of Andrianov L-functions of Siegel cuspforms of degree 2 of weight 3 for the group Γ(4,8), which are contained in [B. van Geemen, D. van Straten, The cuspform of weight 3 on Γ2(2,4,8), Math. Comp. 61 (204) (1993) 849–872]. These L-functions are related to the Galois representations on the Siegel modular threefold Γ(4,8)∖H2 as determined by B. van Geemen and N. Nygaard [B. van Geemen, N.O. Nygaard, On the geometry and arithmetic of some Siegel modular threefolds, J. Number Theory 53 (1995) 45–87].