  We consider the totally asymmetric simple exclusion process (TASEP) on a
periodic one-dimensional lattice of L sites. Using Bethe ansatz, we derive
parametric formulas for the eigenvalues of its generator in the thermodynamic
limit. This allows to study the curve delimiting the edge of the spectrum in
the complex plane. A functional integration over the eigenstates leads to an
expression for the density of eigenvalues in the bulk of the spectrum. The
density vanishes with an exponent 2/5 close to the eigenvalue 0.
