For what number fields $K$ can we actually compute the residue of $\zeta_K(s)$ at the pole $s=1$ directly? Since the class number formula tells us that
this doesn't seem terribly useful unless one can calculate everything except one element. The only examples that I've ever seen involve computing the residue, but I don't think that's too interesting as I would expect that the whole utility of the formula comes from computing the residue and then using whatever we know to deduce either the regulator, class number or discriminant.
The only other utility of the formula that I can come up with is if we have some relationship between the zeta-functions of some number fields. Then one could use the functional equation to compute relations between the regulators, class numbers etc. of these number fields. Picking e.g. an $S_3$ extension we find relationships between the zeta functions of the subextensions.
Could anyone elaborate on if this equation actually has any use in practical computations? Even references to papers where its used to compute something interesting would be appreciated.