Corrections of an elliptic block in the NS sector

We propose a correction to one of the elliptic blocks in the NS sector of 2d $\mathcal{N}=1$ superconformal field theories. We analyze the 4-point block in the pillow geometry to demonstrate the necessity of the correction and verify the formula by numerically checking the crossing symmetries in the $\mathcal{N}=1$ super Liouville theory, as well as directly comparing the $c$-recursion and $h$-recursion results.