Depends on the diode. It can be in the order of 0.1ua or 100ua depending on the zener voltage and where you measure the leakage.
It is generally specified in the data sheet. Download a data sheet for the 1n5221 - 1n5281 family.
1n5221 is 2.4V zener at 20ma with <100uA leakage at 1V
1n5281 is 200V zener at 0.65ma with <0.1uA leakage at 150V.