  We study the spectral zeta functions of the Laplacian on fractal sets which
are locally self-similar fractafolds, in the sense of Strichartz. These
functions are known to meromorphically extend to the entire complex plane, and
the locations of their poles, sometimes referred to as complex dimensions, are
of special interest. We give examples of self-similar sets such that their
complex dimensions are not on the imaginary axis, which allows us to interpret
their Laplacian determinant as the regularized product of their eigenvalues. We
then investigate a connection between the logarithm of the determinant of the
discrete graph Laplacian and the regularized one.
