What are some good references for the Airy functions? In particular, I am interested in the Airy-zeta function

$$\zeta_{Ai}(s)=\sum_{r\in U}\frac{1}{r^s}$$

where $U$ is the set of zeros of $Ai(z)$. I have checked the second reference on the Wolfram page but I'm looking for something more, and the first reference does not seem to be too specific to this function.