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 SiegelKlingen 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 Lfunction 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:
$$\zeta (1k)=\frac{{B}_{k}}{k}$$ where ${B}_{k}$ is the $k$th Bernoulli number;

2.
The values of Dirichlet Lseries at negative integers can be written in terms of generalized Bernoulli numbers^{} as follows:
$$L(1k,\chi )=\frac{{B}_{k,\chi}}{k}$$ where ${B}_{k,\chi}$ is the $k$th generalized Bernoulli number associated to $\chi $.
Therefore:
$$\zeta (1k,K)=\zeta (1k)L(1k,\chi )=\frac{{B}_{k}\cdot {B}_{k,\chi}}{{k}^{2}}.$$ 
The interested reader can find tables of values at the http://www.math.cornell.edu/ alozano/dedekindvalues/index.htmlauthor’s personal website.
Title  values of Dedekind zeta functions of real quadratic number fields at negative integers 

Canonical name  ValuesOfDedekindZetaFunctionsOfRealQuadraticNumberFieldsAtNegativeIntegers 
Date of creation  20130322 16:01:27 
Last modified on  20130322 16:01:27 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  5 
Author  alozano (2414) 
Entry type  Application 
Classification  msc 11R42 
Classification  msc 11M06 
Related topic  FactorizationOfTheDedekindZetaFunctionOfAnAbelianNumberField 