  We obtain formulas for the spectral zeta function of the Laplacian on
symmetric finitely ramified fractals, such as the Sierpinski gasket, and a
fractal Laplacian on the interval. These formulas contain a new type of zeta
function associated with a polynomial (rational functions also can appear in
this context). It is proved that this zeta function has a meromorphic
continuation to a half plain with poles contained in an arithmetic progression.
It is shown as an example that the Riemann zeta function is the zeta functions
of a quadratic polynomial, which is associated with the Laplacian on an
interval. The spectral zeta function of the Sierpinski gasket is a product of
the zeta function of a polynomial and a geometric part; the poles of the former
are canceled by the zeros of the latter. A similar product structure was
discovered by M.L. Lapidus for self-similar fractal strings.
