Let $G \in \text{Gaussian}(0, \sigma)$ be a D210: Gaussian random real number.

Let $Z \in \text{Gaussian}(0, 1)$ be a D211: Standard gaussian random real number. By definition, we have $G \overset{d}{=} \sigma Z$. Result R5409: Expectation of the absolute value of a centred standard gaussian random real number shows that \begin{equation} \mathbb{E} |Z| = \sqrt{\frac{2}{\pi}} \end{equation} Hence, we have \begin{equation} \mathbb{E} |G| = \mathbb{E} |\sigma Z| = \sigma \mathbb{E} |Z| = \sigma \sqrt{\frac{2}{\pi}} \end{equation} $\square$