  We prove a formula, with power savings, for the sixth moment of Dirichlet
L-functions averaged over moduli $q$, over primitive characters $\chi$ modulo
$q$, and over the critical line. Our formula agrees precisely with predictions
motivated by random matrix theory. In particular, the constant 42 appears as a
factor in the leading order term, exactly as is predicted for the sixth moment
of the Riemann zeta-function.
