  We define zeta functions for the adjoint action of GL(n) on its Lie algebra
and study their analytic properties. For n<4 we are able to fully analyse these
functions, and recover the Shintani zeta function for the prehomogeneous vector
space of binary quadratic forms for n=2. Our construction naturally yields a
regularisation, which is necessary for the improvement of the properties of
these zeta function, in particular for the analytic continuation if n>2.
  We further obtain upper and lower bounds on the mean value X^{-5/2}\sum_{E}
\res_{s=1}\zeta_E(s) as X->\infty, where E runs over totally real cubic number
fields whose second successive minimum of the trace form on its ring of
integers is bounded by X. To prove the upper bound we use our new zeta function
for GL(3). These asymptotic bounds are a first step towards a generalisation of
density results obtained by Datskovsky in case of quadratic field extensions.
