  The lowest eigenvalue of the Laplacian within the S-sided regular polygon
with Dirichlet boundary conditions is the focus of this report. As suggested by
others, this eigenvalue may be expressed as an asymptotic expansion in powers
of 1/S where, interestingly, they have shown that the first few coefficients in
that expansion, up to sixth order, may be expressed analytically in terms of
Riemann zeta functions and roots of Bessel functions. This report builds on
that work with three main contributions: (1) compelling numerical evidence
independently supporting those published results, (2) a conjecture adding two
more terms to the asymptotic expansion, and (3) an observation that
higher-order coefficients both alternate in sign and grow rapidly in magnitude,
which suggest the series doesn't converge unless S>=10. This report is based on
a numerical computation of the eigenvalues precise to fifty digits for S up to
150.
  Keywords: Laplacian eigenvalue; regular polygon; asymptotic expansion
