Riemann Siegel theta function

The Riemann Siegel $\vartheta$ function is defined for $t \in \mathbb{R}$ by $$\vartheta(t) = \mathrm{Im} \left( \log \Gamma \left(\dfrac{1}{4}+ i \dfrac{t}{2} \right) - \dfrac{t \log \pi}{2} \right),$$ where $\log$ denotes the logarithm and $\Gamma$ denotes the gamma function.