The "adjusting" of the 34401A is a feasible task, provided you have the references against which you want to make the adjustment.
The "Calibration" menu shows first the password (the standard one is "034401" you may change !! therefore be careful what you do) and deeper in the menu, when you are for example in the DCV 10V range, it shall show you "+10.00000" at the beginning of the adjusting process.
You can adjust a single range only, I did it several times with my DMM (inspired by TiN's video on his xdevs.com).
I did 10V and 1V ranges in my case and it worked fine.
For example - Set the DCV 10V range in your typical setup, wire your reference voltage into the DMM's inputs (it shall be around 10V), go to Calibrate, unlock when necessary, navigate to "+10.00000" on the display, and then with the arrow buttons dial in Your reference voltage (for EXAMPLE "+09.99772"), then press Enter. It will show something like "Full Scale.." on the display during the internal adjustment process running.
Note: you may dial in negative numbers too, therefore take care with the sign..It will do the internal adjusting for some time (like 10-30secs afaik), WAIT (!!), and DO NOT press anything.
After a while it returns to the normal DCV 10V mode automatically and you shall see Your 9.99772 (EXAMPLE only) reference voltage on your display.
Disclaimer: Do it at YOUR OWN RISK, NO WARRANTIES of any kind are provided.. PS: Your 34401A has to be powered on for some time, I did with at least 1 week powered on. The reference voltage shall be also powered on for some time, of course..
Before the adjustment double check your Zero reading is 0V at the 10V range with the above example (with all 4 inputs shorted together properly!)..
For more details on proper wiring, temperatures, etc. do consult the Metrology Section..