This letter presents the first lattice QCD computation of the coupled channel $\pi\Sigma−\bar{K}N$ scattering amplitudes at energies near 1405MeV. These amplitudes contain the resonance $\Lambda(1405)$ with strangeness $S=−1$ and isospin, spin, and parity quantum numbers $I(J^P)=0(1/2^−)$. However, whether there is a single resonance or two nearby resonance poles in this region is controversial theoretically and experimentally. Using single-baryon and meson-baryon operators to extract the finite-volume stationary-state energies to obtain the scattering amplitudes at slightly unphysical quark masses corresponding to $m_\pi\approx200$ MeV and $m_K\approx487$ MeV, this study finds the amplitudes exhibit a virtual bound state below the $\pi\Sigma$ threshold in addition to the established resonance pole just below the $\bar{K}N$ threshold. Several parametrizations of the two-channel K-matrix are employed to fit the lattice QCD results, all of which support the two-pole picture suggested by SU(3) chiral symmetry and unitarity.