  The low-energy neutron$-^{19}$C scattering in a neutron-neutron-core model is
studied with large scattering lengths near the conditions for the appearance of
an Efimov state. We show that the real part of the elastic $s-$wave phase-shift
($\delta_0^R$) presents a zero, or a pole in $ k\cot\delta_0^{R}$, when the
system has an Efimov excited or virtual state. More precisely the pole scales
with the energy of the Efimov state (bound or virtual). We perform calculations
in the limit of large scattering lengths, disregarding the interaction range,
within a renormalized zero-range approach using subtracted equations. It is
also presented a brief discussion of these findings in the context of ultracold
atom physics with tunable scattering lengths.
