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. \[F(t)=∫-∞^∞ {h(x)ζ(4+ix)}{7-t/7}\,dx, Z(t)={ F(t)}{(14+t^2)¹²(25/4+t^2)¹²}.\] After that we get the estimate \[Z(t)=(t/2π)⁷⁴\{eiϑ(t)H(t)\}+O(t-3/4),\] with \[H(t)=∫-∞^∞(t/2π)ix/2ζ(4+it+ix)/7(π x/7)\,dx=(t/2π)⁻⁷⁴∑ₙ₌₁^∞ {1}{n¹²⁺ⁱᵗ}{2}{1+(t/2π n^2)-7/2}.\] We explain how the study of this function can lead to information about the zeros of the zeta function on the critical line.