  We prove that the first Dirichlet eigenvalue of a regular $N$-gon of area
$\pi$ has an asymptotic expansion of the form
$\lambda_1(1+\sum_{n\ge3}C_n(\lambda_1)N^{-n})$ as $N\to\infty$, where
$\lambda_1$ is the first Dirichlet eigenvalue of the unit disk and $C_n$ are
polynomials whose coefficients belong to the space of multiple zeta values of
weight $n$. We also explicitly compute these polynomials for all $n\le14$.
