Alot of e39 will have the sound from belt tensioner. This is mean the tensioner need to change. As long as the belt is tight enough then you no need to to change the belt tensioner.If you able to unplug the belt tensioner then put libricant oil or 4WD oil. Or you can unplug water host in order for you to spray 4WD oil to the bet tensioner.Due to belt tensioner dosent have the bearing, whenever the hole holding the shaft of the belt tensioner is bigger causing the lose of the body. Then the friction between moving parts of the body causing sound. If you able to add lubricant oil to the belt tensioner the sound will reduce.I have changed the new belt tensioner recently. But still got very little sound from the belt tensioner although it is new. To prevent this happen, please add lubricant oil to belt tensioner before fit to your car.