  This paper studies the non-holomorphic Eisenstein series E(z,s) for the
modular surface, and shows that integration with respect to certain
non-negative measures gives meromorphic functions of s that have all their
zeros on the critical line Re(s) = 1/2. For the constant term of the Eisenstein
series it shows that all zeros are on the critical line for fixed y= Im(z) \ge
1, except possibly for two real zeros, which are present if and only if y > 4
\pi e^{-\gamma} = 7.0555+. It shows the Riemann hypothesis holds for all
truncation integrals with truncation parameter T \ge 1. For T=1 this proves the
Riemann hypothesis for a zeta function recently introduced by Lin Weng,
attached to rank 2 semistable lattices over the rationals.
