<A HREF="https://www.jstor.org/stable/2972387
">On the numerical value of $i^i$</A> and <A HREF="https://doi.org/10.2307/2972388">Historical notes on the relation $e^{-\pi/2}=i^i$</A> describe how these accurate computations can be performed with logarithmic tables.

Euler described how he arrived at the identity in a paper read at the Berlin Academy in 1746, giving more decimals (13) than in the letter to Goldbach. A later calculation by Gauss computed 35 decimal places. Euler did not present his computation, but Gauss did [<A HREF="https://archive.org/details/werkecarlf03gausrich/page/n439/mode/2up">source</A>].