Authors’ summary: This paper studies a two-variable zeta function \(Z_K (w, s)\) attached to an algebraic number field \(K\), introduced by G. van der Geer and R. Schoof [Sel. Math., New Ser. 6, No. 4, 377–398 (2000; Zbl 1030.11063)], which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors. When \(w= 1\) this function becomes the completed Dedekind zeta function \(\hat\zeta_K(s)\) of the field \(K\). The function is a meromorphic function of two complex variables with polar divisor \(s(w - s)\), and it satisfies the functional equation \(Z_K(w, s) = Z_K(w,w - s)\). We consider the special case \(K = {\mathbb Q}\), where for \(w = 1\) this function is \(\hat{\zeta}(s)= \pi^{-{s\over 2}} \Gamma({s\over 2}) \zeta(s)\). The function \(\xi_{{\mathbb Q}}(w, s) := {s(s- w)\over 2 w}Z_{{\mathbb Q}}(w, s)\) is shown to be an entire function on \({\mathbb C}^2\), to satisfy the functional equation \(\xi_{{\mathbb Q}}(w, s) = \xi_{{\mathbb Q}}(w, w - s),\) and to have \(\xi_{{\mathbb Q}}(0, s) =-{s^2\over 8}(1 - 2^{1 + {s\over 2}}) (1 - 2^{1 - {s\over 2}}) \hat{\zeta}({s\over 2}) \hat{\zeta}({-s\over 2}).\) We study the location of the zeros of \(Z_{{\mathbb Q}}(w, s)\) for various real values of \(w = u\). For fixed \(u \geq 0\) the zeros are confined to a vertical strip of width at most \(u + 16 \) and the number of zeros \(N_u(T)\) to height \(T\) has similar asymptotics to the Riemann zeta function. For fixed \(u < 0\) these functions are strictly positive on the {“}critical line{”} \({\mathfrak R}(s) = {u\over 2}\). This phenomenon is associated to a positive convolution semigroup with parameter \(u \in {\mathbb R}_{> 0}\), which is a semigroup of infinitely divisible probability distributions, having densities \(P_u(x)dx\) for real \(x\), where \(P_u(x) = {1\over 2\pi}\theta(1)^u Z_{{\mathbb Q}}(-u, -{u\over 2} + ix),\) and \(\theta(1) = \pi^{1/4}/ \Gamma(3/4)\).