  We investigate the photoproduction of $\Lambda(1405)\equiv\Lambda^*$ hyperon
resonance, i.e., $\gamma p\to K^+\Lambda^*$, employing the effective Lagrangian
approach with the $t$-channel Regge trajectories at tree level. We extensively
explore the effects from the nucleon resonances in the vicinity of the
threshold $\sqrt{s}_\mathrm{th}\approx1900$ MeV, i.e., $N(2000)$, $N(2030)$,
$N(2055)$, $N(2095)$, and $N(2100)$, and observe that they are of great
importance to reproduce the recent CLAS/Jlab experimental data. Total and
differential cross sections are given as numerical results and compared with
the experimental data, in addition to the $t$-channel momentum transfer and
photon-beam asymmetry. The invariant-mass plots for $\gamma p\to
K^+\pi^0\Sigma^0$ via $\Lambda^*$ are also extracted from the two-body process
results, showing a qualitative agreement with the data. We also discuss the
constituent-counting rule for the internal structure of $\Lambda^*$, resulting
in that $\Lambda^*$ appears to be different from a simple three-quark $(uds)$
state.
