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 ζ ^ 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 = ℚ , where for w = 1 this function is ζ ^ ( s ) = π - s 2 Γ ( s 2 ) ζ ( s ) . The function ξ ℚ ( w , s ) : = s ( s - w ) 2 w Z ℚ ( w , s ) is shown to be an entire function on ℂ 2 , to satisfy the functional equation ξ ℚ ( w , s ) = ξ ℚ ( w , w - s ) , and to have ξ ℚ ( 0 , s ) = - s 2 8 ( 1 - 2 1 + s 2 ) ( 1 - 2 1 - s 2 ) ζ ^ ( s 2 ) ζ ^ ( - s 2 ) . We study the location of the zeros of Z ℚ ( w , s ) for various real values of w = u . For fixed u ≥ 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" ℜ ( s ) = u 2 . This phenomenon is associated to a positive convolution semigroup with parameter u ∈ ℝ > 0 , which is a semigroup of infinitely divisible probability distributions, having densities P u ( x ) d x for real x , where P u ( x ) = 1 2 π θ ( 1 ) u Z ℚ ( - u , - u 2 + i x ) , and θ ( 1 ) = π 1 / 4 / Γ ( 3 / 4 ) .