Higher Selberg zeta functions (Q1885635)

In [\textit{N. Kurokawa} and \textit{M. Wakayama}, A comparison between the sum over Selberg's zeros and Riemann's zeros. J. Ramanujan Math. Soc. 18, No. 3, 221--236 (2003; Zbl 1052.11062)] the authors introduced a new variant \(z(s)\) of the Selberg zeta function which they call higher Selberg zeta function. In the current paper they show that \(z(s)\) satisfies a functional equation as \(s\) is replaced by \(-s\). They describe the necessary Gamma factor in terms of triple sine functions. The proof relies on the trace formula.