L-functions of S3(Γ2(2,4,8))

van Geemen and van Straten [B. van Geemen, D. van Straten, The cuspform of weight 3 on Γ2(2,4,8), Math. Comp. 61 (1993) 849–872] showed that the space of Siegel modular cusp forms of degree 2 of weight 3 with respect to the so-called Igusa group Γ2(2,4,8) is generated by 6-tuple products of Igusa theta constants, and each of them are Hecke eigenforms. They conjectured that some of these products generate Saito–Kurokawa representations, weak endoscopic lifts, or D-critical representations. In this paper, we prove these conjectures. Additionally, we obtain holomorphic Hermitian modular eigenforms of GU(2,2) of weight 4 from these representations.