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This paper studies a two-variable zeta function $Z_K(w,s)$ attached to an algebraic number field $K$, introduced by van der Geer and Schoof, 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 ${\widehat{\zeta }}_K\left(s\right)$ 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...