  We study structures of the $0^+$ states of $^{13}_\Lambda$C within the
framework of $\alpha+\alpha+\alpha+\Lambda$ four-body orthogonality condition
model with the Gaussian expansion method. We obtain five $0^+$ states of
$^{13}_\Lambda$C and compare them with the low-lying four $0^+$ states of
$^{12}$C. We find that the $0_1^+$, $0_2^+$ and $0_4^+$ states correspond to
the $0_1^+$, $0_2^+$ and $0_4^+$ states of $^{12}$C, respectively. We also find
that both the $0_3^+$ and $0_5^+$ states have analogous nature to the $0_3^+$
state of $^{12}$C, where the additional $\Lambda$ particle couples with $^8$Be
cluster and one $\alpha$ cluster, in the $0_3^+$ state of $^{12}$C,
respectively. The former is then shown to have an elongated
$^9_\Lambda$Be$(0_1^+) + \alpha$ cluster structure with a higher nodal
behaviour for their relative wave function, and the latter have a dilute
three-body gas consisting of $2\alpha+^5_\Lambda$He clusters.
