oil service and inspection lights

I'm working on a 1985 325 bmw. This a new car for me and I just have a coulple of general questions. Are the oil service lights and inspection lights supposed to stay on when the ignition is turned off? I have tried to reset them with no luck. Is this a problem with the batteries on the panel?

Thanks

that is what they say goes bad, but when you have it apart I would just pull out the little board with the lights on.. and stop at that..