Approximate formula for $Z(t)$

The series for the zeta function does not converge on the critical line but the function $$G(t)=\sum_{n=1}^\infty \frac{1}{n^{\frac12+it}}\frac{t}{2\pi n^2+t}$$ satisfies $Z(t)=2\Re\{e^{i\vartheta(t)}G(t)\}+O(t^{-\frac56+\varepsilon})$. So one expects that the zeros of zeta on the critical line are very near the zeros of $\Re\{e^{i\vartheta(t)}G(t)\}$. There is a related function $U(t)$ that satisfies the equality $Z(t)=2\Re\{e^{i\vartheta(t)}U(t)\}$.