  We investigate the order $\rho$ of the four entire functions in the
Nevanlinna matrix of an indeterminate Hamburger moment sequence. We give an
upper estimate for $\rho$ which is explicit in terms of the parameters of the
canonical system associated with the moment sequence via its three-term
recurrence. Under a weak regularity assumption this estimate coincides with a
lower estimate, and hence $\rho$ becomes computable. Dropping the regularity
assumption leads to examples where upper and lower bounds do not coincide and
differ from the order. In particular we provide examples for which the order is
different from its lower estimate due to M.S.Liv\v{s}ic.
