Hi Jacques,
I'm working on a 34401a repair at the moment. I'm think it is likely that your display is fine, or at least not the only problem, as when I remove the display on my 34401a it still beeps. With a working display attached it gives one or two short beeps, and with no display it gives 2 long beeps.
I would suspect an issue with the mainboard floating logic. Perhaps a good place to start (after the obvious power supply stuff) is the XPONRST signal (power good) from pin 5 of U553. Can't really add much more as my problems are all analogue related - my floating logic section seems to be working fine. Also might be worth scoping out the signals on the optocouplers (U596, U704) to see if the ground referenced logic is talking to the floating logic.
Good luck!
edit: the LM337 on this board is a variant of the SMD D2PAK TO-263 package. These dont normally have the mounting hole though. You can easily buy the D2PAK variant without mounting hole, but I bent the legs of a TO-220 to save buying more parts.