Special values of Dedekind zeta functions can frequently be expressed (and then calculated) in topological terms, e.g. from the Euler characteristics of certain manifolds/orbifolds.

An easy-to-formulate example is that of zeta functions of totally real quadratic fields $K$: the value of $\zeta_K(2)$ can be computed from the orbifold Euler characteristics of the Hilbert modular surface $$SL(2,O_K)\backslash (H^2\times H^2),$$
where $SL_2(O_K)$ is embedded into $SL(2,R)\times SL(2,R)\subset Isom^+(H^2\times H^2)$ via the two different embeddings $K\to R$.

More involved examples can be found in http://people.mpim-bonn.mpg.de/zagier/files/scanned/ValuesofZetaFunsAndApplications/fulltext.pdf and its references.

In a similar vein, the class number of $O_K$ equals the number of cusps (= the number of ends) of the Hilbert modular surface, which (I hope) might be easier to compute.