  The principal aim in this paper is to employ a recently developed unified
approach to the computation of traces of resolvents and $\zeta$-functions to
efficiently compute values of spectral $\zeta$-functions at positive integers
associated to regular (three-coefficient) self-adjoint Sturm--Liouville
differential expressions $\tau$. Depending on the underlying boundary
conditions, we express the $\zeta$-function values in terms of a fundamental
system of solutions of $\tau y = z y$ and their expansions about the spectral
point $z=0$. Furthermore, we give the full analytic continuation of the
$\zeta$-function through a Liouville transformation and provide an explicit
expression for the $\zeta$-regularized functional determinant in terms of a
particular set of this fundamental system of solutions.
  An array of examples illustrating the applicability of these methods is
provided, including regular Schr\"{o}dinger operators with zero, piecewise
constant, and a linear potential on a compact interval.
