Integral Representation for Riemann-Siegel $Z(t)$ function

We apply Poisson formula for a strip to give a representation of $Z(t)$ by means of an integral. $$F(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{h(x)\zeta (4+ix)}{7\cosh \pi \frac{x-t}{7}}dx,Z(t)=\frac{\mathrm{\Re }F(t)}{(\frac{1}{4}+t^2{)}^{\frac{1}{2}}(\frac{25}{4}+t^2{)}^{\frac{1}{2}}}\mathrm{.}$$ After that we get the estimate $$Z(t)=\left(\frac{t}{2\pi }{)}^{\frac{7}{4}}\mathrm{\Re }\right\{e^{i\vartheta (t)}H(t)\}+O(t^{-3\mathrm{/}4}),$$ with $$H(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(\frac{t}{2\pi }{)}^{ix\mathrm{/}2}\frac{\zeta (4+it+ix)}{7\cosh (\pi x\mathrm{/}7)}dx=(\frac{t}{2\pi }{)}^{-\frac{7}{4}}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{\frac{1}{2}+it}}\frac{2}{1+(\frac{t}{2\pi n^2}{)}^{-7\mathrm{/}2}}\mathrm{.}$$ We explain how the study of this function can lead to information about the zeros of the zeta function on the critical line.