Note on the Selberg trace formula for the Picard group (Q1057912)

In his version of the Selberg trace formula for the Picard group, \textit{J. Szmidt} [Acta Arith. 42, 391--424 (1983; Zbl 0539.10024)] introduced an analogue \(\gamma_{\mathbb{Q}(i)}\) of the Euler constant for the Gaussian number field \(\mathbb{Q}(i)\) by means of the zeroth Laurent coefficient of the Dedekind zeta-function of \(\mathbb{Q}(i)\) at \(s=1\). In the present note it is proved that \(\gamma_{\mathbb{Q}(i)}\) can be expressed in terms of the Euler constant \(\gamma\), \(log \Gamma(1/4)\), and \(\pi\), \(\log 2\), \(\log \pi\). An analogous result holds for all imaginary quadratic number fields of class-number one.