Expected payoff of an at-the-money call with normal prices

Suppose the stock price at expiry is normally distributed with mean equal to today's price $S_0$ and standard deviation $\sigma$. What is the expected payoff of a call struck at $S_0$?

$\sigma/\sqrt{2\pi}\approx0.4\,\sigma$. The payoff is $\sigma Z^+$ with $Z$ standard normal, and $\mathbb E[Z^+]=\int_0^\infty z\,\varphi(z)\,dz=\tfrac{1}{\sqrt{2\pi}}$.