Using Mathematica's Asymptotic function I get the following identity. For $s\approx 0$ we have
$$\left(\frac{s}{2}+\frac{5 \sqrt{\frac{2}{3}} \sqrt{s}}{2 \tan ^{-1}\left(\frac{\sqrt{\frac{2}{3}}}{\sqrt{s}}\right)}\right)^{-1} \approx \sqrt{\frac{3\pi^2}{50 s}}$$
The simplified formula fits my data really well (see discussion), any idea where this result comes from?
Is there a standard series expansion where the leading term is of the form $a_0 s^{-\frac{1}{2}}$?