Green's function of the three-dimensional convective Helmholtz equation for an infinite straight pipe of arbitrary (but constant along its length) cross-sectional shape and area, having either acoustically rigid or acoustically soft walls or the walls of a mixed type, is constructed. This function is represented by a series in pipe acoustic modes. In the function, the effects of a uniform mean flow in the pipe are directly reflected. The effects become more significant, as the flow Mach number increases, and cause, in particular, the appearance and a further growth of the function asymmetry relative to the pipe cross-section, in which the acoustic source is located. Vice versa, the decrease of the Mach number results in the decrease of the effects and, in particular, the decrease of the indicated function asymmetry. In the absence of a flow, the obtained Green function is symmetric with respect to this cross-section.