central locking is dubious on mine (to say the least, you have to give the key a FIRM turn, and even then the passenger front door and the boot dont join in!
..mine also has an alarm, and a fob, which lights up when you press the button, but doesnt seem to do anything... so like a true hero i decided to remove it....
unplugged the main unit, and spotted the "caution - landrover alarm fitted" sticker on the window....
i tried the ignition and nowt happened...
so, going to strip down the doors, i suspect the actuator on the passenger side is worn out, and the switch gear on the drivers side is also worn.
i imagine this is faierly common with a lot of vehicles, so just a case of open and look!
hope this helps!