  The large-distance behavior of the density-density correlation function in
the Lieb-Liniger model at finite temperature is investigated by means of the
recently derived nonlinear integral equations characterizing the correlation
lengths. We present extensive numerical results covering all the physical
regimes fromweak to strong interaction and all temperatures. We find that the
leading term of the asymptotic expansion becomes oscillatory at a critical
temperature which decreases with the strength of the interaction. As we
approach the Tonks-Girardeau limit the asymptotic behavior becomes more complex
with a double crossover of the largest and next-largest correlation lengths.
The crossovers exist only for intermediate couplings and vanish for $\gamma=0$
and $\gamma=\infty$.
