  We perform a coupled-channels study of the low-lying states in
$^{13,15,17,19}_{~~~~~~~~~~~~~~~~~~~~~\Lambda}$C with a covariant energy
density functional based microscopic particle-core coupling model. The energy
differences of $1/2^-$ and $3/2^-$ states in $^{13}_\Lambda$C and
$^{15}_\Lambda$C are predicted to be 0.25 MeV and 0.34 MeV, respectively. We
find that configuration mixings in the $1/2^-$ and $3/2^-$ states of
$^{15}_\Lambda$C are the weakest among those of
$^{13,15,17,19}_{~~~~~~~~~~~~~~~~~~~~~\Lambda}$C. It indicates that
$^{15}_\Lambda$C provides the best candidate among the carbon hypernuclei to
study the spin-orbit splitting of $p_\Lambda$ hyperon state.
