Levinson Functions

Abstract: Starting from some of Norman Levinson's results, we construct interesting examples of functions $f(s)$ such that for $s=\frac12+it$, we have $Z(t)=2\Re{\pi^{-\frac{s}{2}}\Gamma(s/2)f(s)}$. For example one such function is [\begin{aligned}{\mathcal R }{-3}(s)=\frac12&\int{0\swarrow1}\frac{x^{-s}e^{3\pi ix^2}}{e^{\pi i x}-e^{-\pi i x}}\,dx\&+\frac{1}{2\sqrt{3}}\int_{0\swarrow1}\frac{x^{-s}e^{\frac{\pi i}{3}x^2}}{e^{\pi i x}-e^{-\pi i x}}\Bigl(e^{\frac{\pi i}{2}}+2e^{-\frac{\pi i}{6}}\cos(\tfrac{2\pi x}{3})\Bigr)\,dx.\end{aligned}]