Probability Laws Concerning Zeta Integrals

We give a probabilistic interpretation of the Dedekind zeta functions of $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-2})$ using zeta integrals and use this to show that the first two Li coefficients of these zeta functions are positive. This extends a result of Biane, Pitman, and Yor (2001) which considered the case of the Riemann zeta function.