  This paper discusses the simplest examples of spectral zeta functions,
especially those associated with graphs, a subject which has not been much
studied. The analogy and the similar structure of these functions, such as
their parallel definition in terms of the heat kernel and their functional
equations, are emphasized. Another theme is to point out various contexts in
which these non-classical zeta functions appear. This includes Eisenstein
series, the Langlands program, Verlinde formulas, Riemann hypotheses, Catalan
numbers, Dedekind sums, and hypergeometric functions. Several open-ended
problems are suggested with the hope of stimulating further research.
