An expression for Riemann Siegel function

There are many analytic functions $U(t)$ satisfying $Z(t)=2\mathrm{\Re }\{e^{i\vartheta (t)}U(t)\}$. Here, we consider an entire function $(s)$ such that $U(t)=(\frac{1}{2}+it)$ is one of the simplest among them. We obtain an expression for the Riemann-Siegel function $Z(t)$ in terms of the zeros of $(s)$. Implicitly, the function $(s)$ is considered by Riemann in his paper on Number Theory. Riemann spoke of having used an expression for $\mathrm{\Xi }(t)$ in his demonstration that most of the non-trivial zeros of the zeta function lie on the critical line. Therefore, any expression deserves a study.