Ok, the seller told me that the 34401A was a backup DMM for his lab, but he never used it. It was made in 2000, so if it really was not used it was not used for 17+ years! The DMM looks absolutely new.
So, should I just leave it powered on for the next days or weeks?
The correct question, what do you want to measure, or how stable do you want to measure?
The 34401A is stable to a few tenths of a ppm, if the RT does not change.
But you also need a superior voltage reference to detect that.
As this is well below the specification limit of this instrument (about 20ppm ball park), don't waste money for energy and lifetime of the DMM.
The instrument seems to stabilize well within 2h to a few ppm, nothing more should be expected.
Also I doubt, that the LM399 reference shows such a big hysteresis from idle state, especially compared to the specification.
Frank