We report on an efficient numerical technique for directly locating transmission resonances and zeros in semiconductor heterostructures using tight-binding multiband models. The quantum transmitting boundary method is employed to generate the inverse of the retarded Green's function ${\mathit{G}}^{\mathit{R}}$(E) in the tight-binding representation. The poles of ${\mathit{G}}^{\mathit{R}}$(E) are located by solving a nonlinear non-Hermitian eigenvalue problem. The eigenvalues are calculated using a shift and invert nonsymmetric Lanczos algorithm followed by Newton refinement. We demonstrate that resonance line shapes are accurately characterized by the location of the poles and zeros of ${\mathit{G}}^{\mathit{R}}$(E) in the complex energy plane. The real part of the pole energy corresponds to the resonance peak and the imaginary part corresponds to the resonance width. A Fano resonance is characterized by a zero-pole pair in the complex energy plane. In the case of an isolated Fano resonance, the zero always occurs on the real energy axis. However, we demonstrate that for overlapping Fano resonances the zeros can move off of the real axis in complex conjugate pairs. This behavior is examined using a simple analytic model for multichannel scattering.