In the paper *On Epstein's zeta-function*, Chowla and Selberg give a formula for evaluating the Dedekind eta function
$$\eta (\tau)=e^{\pi i\tau/12}\prod_{n=1}^\infty (1-e^{2\pi i n\tau}),\quad \Im\tau\gt 0$$
when its argument is imaginary quadratic; it has a closed form in terms of the gamma function at rational arguments.

In turn, this gives the normalized Eisenstein series
$$E_{2k}(\tau)=1-\frac{4k}{B_{2k}}\sum_{n=1}^\infty \sigma_{2k-1}(n)e^{2\pi i\tau n}$$
($\sigma$ is the divisor function, $B$'s are Bernoulli numbers) at imaginary quadratic arguments in terms of the gamma function at rational arguments **for $k\ge 2$ only** – meaning $E_2$ is left out.

This left me wondering: How to evaluate $E_2$ at imaginary quadratic arguments in terms of the gamma function at rational arguments?

It is known for $\tau=i\sqrt{r}$ where $r$ is positive rational, as suggested by the alpha function in *Pi and the AGM* (p. 152, p. 164 Ex. 15) by Borwein & Borwein.

But how should I evaluate, for example, $E_2\left(\frac{1+5i}{2}\right)$?