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Álvaro Lozano-Robledo
alozano@math.cornell.edu
H.C. Wang Assistant Professor at Cornell University
Office: 584 Malott Hall

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Values of Dedekind zeta functions at Negative Integers.

Let K be a totally real number field and let Z(s,K) be the Dedekind zeta function of K. By the Siegel-Klingen Theorem, if n > 0 then Z(-n,K) is a rational number. In this page we provide some tables with these rational values. When K is a real quadratic number field, Z(-n,K) is easily calculated because Z(s,K) factors as Z(s,K)=L(s)L(s,Chi) where L(s) is the Riemann Zeta function and L(s,Chi) is the Dirichlet L-series associated to the quadratic Dirichlet character Chi (of conductor N). Moreover, L(1-k)=-B_k/k (where B_k is the kth Bernoulli number) and L(1-k,Chi) = -B_{k,Chi}/k (where B_{k,Chi} is the generalized Bernoulli number associated to Chi). Thus Z(1-k,K)= B_k B_{k,Chi}/k^2. For more details click here.

Real Quadratic Number Fields:

Let K be a real quadratic number field of discriminant N. Below you can find PARI files (simply type \r cl1-200.gp in GP calculator) for Z(-j,K) for j=1,...,200 and the following quadratic number fields:

cl1-200.gp contains the values for the first 59 real quadratic number fields ("rqnf" in short) of class number 1 (as they appear in Sloane's sequence A003656).
cl2-200.gp contains the values for the first 52 rqnf of class number 2 ( A094619 ).
cl3-200.gp contains the values for the first 42 rqnf of class number 3 ( A094612 ).

The values were calculated with PARI using the routines in the file genbern.gp, also including programs to calculate a) Generalized Bernoulli numbers for quadratic Dirichlet characters modulo N, b) Values of Dirichlet L-functions for quadratic Dirichlet characters modulo N, c) Values of Dedekind Zeta Functions for real quadratic number fields K=Q(\sqrt{N}).

For example, if we type:
gp> \r cl1-200.gp
gp> Z5[6]
%1 = 67/360
Here Z5[6]=Z(1-6,K)=Z(-5,K) where K=Q(\sqrt{5}). If we type N, Pari displays a vector with the first 59 discriminants of real quadratic number fields of class number 1. If N is any of these discriminants, then ZN[k] returns the rational value Z(1-k,K) where K=Q(\sqrt{N}).

References:

Some tables of values of Dedekind zeta functions, by Eyal Goren.
A PARI package to compute values of motivic L-functions, by Tim Dokchitser. The theoretical background can be found in his article "Computing Special Values of Motivic L-functions" in Experimental Mathematics.
See also the following article by Louboutin, Stéphane, Numerical evaluation at negative integers of the Dedekind zeta functions of totally real cubic number fields. Algorithmic number theory, 318--326, Lecture Notes in Comput. Sci., 3076,
Springer, Berlin, 2004.
See also the following article by Stavros Garoufalidis and James E. Pommersheim, Values of Zeta Functions at Negative Integers, Dedekind Sums and Toric Geometry. Journal of the American Mathematical Society, Volume 14, Number 1, Pages 1-23.