Levinson Functions

Starting from some of Norman Levinson’s results, we construct interesting examples of functions $f(s)$ such that for $s=\frac{1}{2}+it$, we have $Z(t)=2\mathrm{\Re }\{{\pi }^{-\frac{s}{2}}\mathrm{\Gamma }(s\mathrm{/}2)f(s)\}$. For example one such function is $$\begin{array}{rl}{\displaystyle {\mathcal{R}}_{-3}(s)=\frac{1}{2}}& {\displaystyle {\int }_{0\swarrow 1}\frac{x^{-s}{e}^{3\pi i{x}^2}}{e^{\pi ix}-e^{-\pi ix}}dx}\\ {\displaystyle }& {\displaystyle +\frac{1}{2\sqrt{3}}{\int }_{0\swarrow 1}\frac{x^{-s}{e}^{\frac{\pi i}{3}{x}^2}}{e^{\pi ix}-e^{-\pi ix}}(e^{\frac{\pi i}{2}}+2e^{-\frac{\pi i}{6}}\cos (\frac{2\pi x}{3}))dx\mathrm{.}}\end{array}$$