RiemannSiegelThetaFunction

The Riemann-Siegel theta function, denoted as $\vartheta(t)$, is an odd real analytic function for real values of $t$ with roots at $0$ and $\pm 17.8455995405 \ldots$. It is an increasing function for $|t| > 6.29$ and has local extrema at $\pm 6.289835988 \ldots$, with value $\mp 3.530972829 \ldots$. The function has a single inflection point at $t = 0$ with

$$\vartheta'(0) = -\frac{\ln \pi + \gamma + \frac{\pi}{2} + 3\ln 2}{2} = -2.6860917 \ldots$$

where $\gamma$ is the Euler-Mascheroni constant and $\vartheta'(0)$ represents the derivative of $\vartheta(t)$ at $t = 0$. This value is the minimum of the derivative of the function.