  In this article, we study the asymptotics of Dirichlet eigenvalues and
eigenfunctions of the fractional Laplacian $(-\Delta)^s$ in bounded open
Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see
that all eigenvalues converge to $1$ as $s \to 0^+$, we show that the first
order correction in these asymptotics is given by the eigenvalues of the
logarithmic Laplacian operator, i.e., the singular integral operator with
symbol $2\log|\xi|$. By this we generalize a result of Chen and the third
author which was restricted to the principal eigenvalue. Moreover, we show that
$L^2$-normalized Dirichlet eigenfunctions of $(-\Delta)^s$ corresponding to the
$k$-th eigenvalue are uniformly bounded and converge to the set of
$L^2$-normalized eigenvalues of the logarithmic Laplacian. In order to derive
these spectral asymptotics, we need to establish new uniform regularity and
boundary decay estimates for Dirichlet eigenfunctions for the fractional
Laplacian. As a byproduct, we also obtain corresponding regularity properties
of eigenfunctions of the logarithmic Laplacian.
