The onset of longitudinal vortex rolls due to buoyancy forces in the thermal entrance region of plane Poiseuille flow heated from below with a constant heat flux is investigated theoretically. In the present study, the Boussinesq equations are solved numerically by using the finite volume method to examine this roll-type convection. In order to observe the behavior of the vortex rolls aligned along the streamwise position x, the local growth rates of mean fields and fluctuations, r0 and r1, are defined. We suggest that the critical condition of the onset of convective instability would be r0= r1. Also, we clarify the characteristic positions for the onset of convective motion and the undershoot of Nusselt number as well as the above intrinsic instability in comparison with available experimental data.