Define the [Airy zeta function][1] for $n=2,3,\dots$, by
$$
Z(n) := \sum_r \frac{1}{r^n}.
$$
where the sum is over the zeros $r$ of the [Airy function][2] $\operatorname{Ai}$.

In *Mathematica* the $\operatorname{Ai}$ function is implemented as [`AiryAi`][3], and the zeros of this function is implemented as [`AiryAiZero`][4].

I have tried to calculate the values of $Z$ using the following:
`Z[n_] := Sum[1/AiryAiZero[k]^n, {k, 1, Infinity}]`, and then `N[Z[2]]` for example.

Sadly, in *Mathematica 9.0* it gives $0.499$; however, the correct result is $0.531457$. For larger $n$ values $Z$ is correct; however, for only a few digits, even if I modify `$MaxExtraPrecision`.

The Airy zeta function MathWorld page gives a closed-form of $Z$, but then you have to implement $n$th derivatives of different Airy-related functions.

> How do I implement an efficient $Z$ function?


  [1]: http://mathworld.wolfram.com/AiryZetaFunction.html
  [2]: http://mathworld.wolfram.com/AiryFunctions.html
  [3]: https://reference.wolfram.com/language/ref/AiryAi.html
  [4]: https://reference.wolfram.com/language/ref/AiryAiZero.html