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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. 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.