values of Dedekind zeta functions of real quadratic number fields at negative integers

Let $K$ be a real quadratic number field of discriminant $D_K$ and let $\zeta (s,K)$ be the Dedekind zeta function associated to $K$. By the Siegel-Klingen Theorem, if $n>0$ then $\zeta (-n,K)$ is a rational number. On the other hand, $K$ is obviously an abelian number field, thus the factorization of the Dedekind zeta function of an abelian number field tells us that:

$$\zeta (s,K)=\zeta (s)L(s,\chi )$$

where $\zeta (s)$ is the famous Riemann zeta function and $L(s,\chi )$ is the Dirichlet L-function associated to $\chi$, where $\chi$ is the unique Dirichlet character with conductor $D_K$ such that the group of characters of $K/\mathbb{Q}$ is $\{{\chi }_0,\chi \}$ and ${\chi }_0$ is the trivial character. In fact, the values of $\chi$ are simply given by

$$\chi (a)=\left(\frac{D_K}{a}\right)$$

where the parentheses denote the Kronecker symbol.

Furthermore, if $k$ is a positive integer then:

1. Putting the values of the Riemann zeta function in terms of Bernoulli numbers one gets:

   $$\zeta (1-k)=-\frac{B_k}{k}$$

   where $B_k$ is the $k$th Bernoulli number;

2. The values of Dirichlet L-series at negative integers can be written in terms of generalized Bernoulli numbers as follows:

   $$L(1-k,\chi )=-\frac{B_{k,\chi }}{k}$$

   where $B_{k,\chi }$ is the $k$th generalized Bernoulli number associated to $\chi$.

Therefore:

$$\zeta (1-k,K)=\zeta (1-k)L(1-k,\chi )=\frac{B_k\cdot B_{k,\chi }}{k^2}.$$

The interested reader can find tables of values at the http://www.math.cornell.edu/ alozano/dedekind-values/index.htmlauthor’s personal website.