  Recently there has been a renewed interest in the spectra and role in
dynamical properties of excited states of the spin-1/2 Heisenberg
antiferromagnetic chain in longitudinal magnetic fields associated with Bethe
strings. The latter are bound states of elementary magnetic excitations
described by Bethe-ansatz complex non-real rapidities. Previous studies on this
problem referred to finite-size systems. Here we consider the thermodynamic
limit and study it for the isotropic spin-1/2 Heisenberg XXX chain in a
longitudinal magnetic field. We confirm that also in that limit the most
significant spectral weight contribution from Bethe strings leads to gapped
continua in the spectra of the spin +- and xx dynamical structure factors. The
contribution of Bethe strings to the zz dynamical structure factor is found to
be small at low spin densities and to become negligible upon increasing that
density above 0.317. For the -+ dynamical structure factor, that contribution
is found to be negligible at finite magnetic field. We derive analytical
expressions for the line shapes of the +-, xx, and zz dynamical structure
factors valid in the vicinity of singularities located at and just above the
gapped lower thresholds of the Bethe-string states's spectra. As a side result
and in order to provide an overall physical picture that includes the relative
location of all spectra with a significant amount of spectral weight, we
revisit the general problem of the line-shape of the transverse and
longitudinal spin dynamical structure factors at finite magnetic field and
excitation energies in the vicinity of other singularities. This includes those
located at and just above the lower thresholds of the spectra that stem from
excited states described by only real Bethe-ansatz rapidities.
