  We provide a complete characterization of those non-elliptic semigroups of
holomorphic self-maps of the unit disc for which the linear span of
eigenvectors of the generator of the corresponding semigroup of composition
operators is weak-star dense in $H^\infty$. We also give some necessary and
sufficient conditions for completeness in $H^p$. This problem is equivalent to
the completeness of the corresponding exponential functions in $H^\infty$ (in
the weak-star sense) or in $H^p$ of the Koenigs domain of the semigroup. As a
tool needed for the results, we introduce and study discontinuities of
semigroups of holomorphic self-maps of the unit disc.
