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The round circle with the Z is the power LED. It should be on whenever the system is powered up. Going to the left, the next LED is the Caps Lock indicator (upward pointing arrow with an A in the middle). It will turn on for a couple of seconds when power is applied and then go out. To the left of that is the Num Lock indicator (upward arrow with a 1). It will also turn on for a couple of seconds and then go out. The fourth LED to the left of the power button is the Hard Disk access light. It should
briefly flash on and then off, before the Caps Lock and Num Lock lights go out. This is the normal procedure.
Any beeps which occur during power up usually indicates a problem with the system. The exact number, the length of each beep (a short versus a long one) and any pauses in the beep sequence will tell you (in most cases) what type of problem has been detected. A table of beep symptoms and their meanings can be found in the Hardware Maintenance Manual (HMM). The one for the X31 is here:
http://www-307.ibm.com/pc/support/site. ... MIGR-44018 (an Adobe PDF file)
There does not seem to be any error condition that results in exactly six beeps (see page 68 of the HMM). The closest ones are five short beeps (system board problem) or the eight beeps (memory error). The eight beeps occur in a specific sequence which is usually named "1-3-3-1" here in the forums. That stands for one short beep (1), a pause (-), three short beeps (3), another pause (-), three more beeps (3), pause (-) and a final beep (1).
Given that your battery has gone dead, I can suggest that you remove it and try powering up the system using just the AC adapter. If the system operates normally, then you could try charging the battery for a couple of hours with the laptop off and see if that solves your problem. If not, then you may need a new battery. My own X31 has almost the same problem. If the battery is installed, pressing the power button starts the system up but only for about 1/2 second. The Power, Caps Lock and Num Lock LEDs flash, the fan spins but the system shuts off. However I am not getting any beep errors. If the battery is removed, then the X31 boots and operates fine. I know that the battery is dead (measures near 0V) but I haven't gone out and replaced it yet.