Airy zeta function

The Airy function $\mathrm{Ai}$ is oscillatory for negative values of $x$. This yields a sequence of zeros $\{a_i\}_{i=1}^{\infty}$. We define the Airy zeta function using these zeros in the following way: $$\zeta_{\mathrm{Ai}}(z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{|a_k|^z}.$$ It can be shown that $$\zeta_{\mathrm{Ai}}(2) = \dfrac{3^{\frac{5}{3}}\Gamma^4(\frac{2}{3})}{4\pi^2}.$$

References

Airy zeta function (Wikipedia) Airy zeta function (Mathworld)