  As a generalization of the Dedekind zeta function, Weng defined the high rank
zeta functions and proved that they have standard properties of zeta functions,
namely, meromorphic continuation, functional equation, and having only two
simple poles. The rank one zeta function is the Dedekind zeta function. For the
rank two case, the Riemann hypothesis is proved for a general number field.
Recently, he defined more general new zeta function associated to a pair of
reductive group and its maximal parabolic subgroup. As well as high rank zeta
functions, the new zeta function satisfies standard properties of zeta
functions.In this paper, we prove that the Riemann hypothesis of Weng's zeta
function attached to the sympletic group of degree four.This paper includes an
appendix written by L. Weng, in which he explains a general construction for
zeta functions associated to $Sp(2n)$.
