  In this paper, the reaction of electron-positron annihilation into
$\Lambda_c^+\bar{\Lambda}_c^-$ is investigated. The
$\Lambda_c^+\bar{\Lambda}_c^-$ scattering amplitudes are obtained by solving
the Lippmann-Schwinger equation. The contact, annihilation, and two
pseudoscalar-exchange potentials are taken into account in the spirit of the
chiral effective field theory. The amplitudes of $e^+e^-\to
\Lambda_c^+\bar{\Lambda}_c^-$ are constructed by the distorted wave Born
approximation method, with the final state interactions of the
$\Lambda_c^+\bar{\Lambda}_c^-$ re-scattering implemented. By fitting to the
experimental data, the unknown couplings are fixed, and high-quality solutions
are obtained. With these amplitudes, the individual electromagnetic form
factors in the timelike region, $G_E^{\Lambda_c}$, $G_M^{\Lambda_c}$, and their
ratio, $G_E^{\Lambda_c}/G_M^{\Lambda_c}$, are extracted. Both modulus and
phases are predicted. These individual electromagnetic form factors reveal new
insights into the properties of the $\Lambda_c$. The separated contributions of
the Born term, contact, annihilation, as well as the two pseudoscalar exchange
potentials to the electromagnetic form factors are isolated. It is found that
the Born term dominates the whole energy region. The contact term plays a
crucial role in the enhancement near the threshold, and the annihilation term
is essential in generating the fluctuation of the electromagnetic form factors.
