Explicit identities on zeta values over imaginary quadratic fields

Abstract

In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta function at any even integer over any totally real number field are quite well known in literature. We here exhibit the identities for both even and odd values of the Dedekind zeta function over an imaginary quadratic field which are analogous to Ramanujan's identities for even and odd zeta values over $Q$. Moreover, any complex zeta values over imaginary quadratic field may also be evaluated from our identities.