The idea is to iterate on the integral equation starting with $z(s)=1$. This will give the integrals in a series
$$
   I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty
}^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {
(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos {(s_{2n-1}^{2}-s_{2n}^{2})}.
$$
The paper discussing them is [this one][1]. The general formula is
$$
   I_n=\frac{2}{n!}\left(\frac{\pi}{4}\right)^n
$$
and you should take into account also powers of $\gamma$. These are the terms of the power series of the exponential multiplied from a factor 2.


  [1]: http://arxiv.org/abs/1201.1975