We report new experimental results of the oscillatory flow of a solution of giant micelles in a narrow vertical tube. We explore small driving amplitudes (laminar flow) and a large range of driving frequencies. High-resolution particle image velocimetry measurements of the flow field in a meridional plane of the tube show that the velocity magnitude at the tube axis peaks at well-defined resonance frequencies, where the phase lag with the forcing changes abruptly. The velocity field is highly inflectional and eventually exhibits some hints of shear banding. The results are compared with theoretical predictions based on the upper-convected Maxwell and Oldroyd-B models of the linear shear rheology of the solution, and observed coincidences and discrepancies are discussed in detail.