Airy zeta function at 2

Theorem

The following formula holds: $$\zeta_{\mathrm{Ai}}(2)=\dfrac{3^{\frac{5}{3}}\Gamma^4(\frac{2}{3})}{4\pi^2},$$ where $\zeta_{\mathrm{Ai}}$ denotes the Airy zeta function, $\Gamma$ denotes the gamma function, and $\pi$ denotes pi.