  The Riemann zeta function $\zeta(s)$ is defined as the infinite sum
$\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann
hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line
${\rm Re}\,s= \frac{1}{2}$. Thus, to find these zeros it is necessary to
perform an analytic continuation to a region of complex $s$ for which the
defining sum does not converge. This analytic continuation is ordinarily
performed by using a functional equation. In this paper it is argued that one
can investigate some properties of the Riemann zeta function in the region
${\rm Re}\,s<1$ by allowing operator-valued zeta functions to act on test
functions. As an illustration, it is shown that the locations of the trivial
zeros can be determined purely from a Fourier series, without relying on an
explicit analytic continuation of the functional equation satisfied by
$\zeta(s)$.
