24V H-Bridge DC motor, why voltage drop to 3v if motor connected
Now I found why it's not working... Well, actually it works if and only if it start with logic 00 (undriven state) in it's input pins. Any input pin can only be pulled high after some brief period of time.

It is suggested in L298N datasheet (in Application Information section, 2nd point, last paragraph -- page 7 of 14) that "Before to Turn-ON the Supply Voltage and before to Turn it OFF, the Enable input must be driven to the Low state."

Since the enable line is boolean OR product of the logic inputs, both of them must be set to zero (low) before supply applied to Vs (power supply).