Prove that following integral function has only real zeros.

Question

I have downloaded a PDF about Riemann $\xi$ function.

(https://projecteuclid.org/euclid.acta/1485887676)

There is written (in German) that following function has only real zeros:

$$\xi^*(z)=8\pi^2\int_{0}^{\infty}(e^{\frac{9u}{2}}+e^{-\frac{9u}{2}})e^{-\pi(e^{2u}+e^{-2u})}\cos(zu)du$$

However i don't understand german enough to see the proof of this assertion.

Please show me how to prove that $\xi^{*}$ has only real zeros or at least point where the proof is written.

Regards