Approximate formula for $Z(t)$

The series for the zeta function does not converge on the critical line but the function $$G(t)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{\frac{1}{2}+it}}\frac{t}{2\pi n^2+t}$$ satisfies $Z(t)=2\mathrm{\Re }\{e^{i\vartheta (t)}G(t)\}+O(t^{-\frac{5}{6}+\epsilon })$. So one expects that the zeros of zeta on the critical line are very near the zeros of $\mathrm{\Re }\{e^{i\vartheta (t)}G(t)\}$. There is a related function $U(t)$ that satisfies the equality $Z(t)=2\mathrm{\Re }\{e^{i\vartheta (t)}U(t)\}$.