You can use a White and HB Red LED in series to make a 5.1 V zener too. ESR is around 15 Ohm @ 100mW , so 2 LEDs in series ~ 30 Ohms (+/- 50% typ) at 20 mA
Thus is bias current changes +/-10% voltage changes 2mA * 30 Ω = 60mV
or a 0.06/5.1 = for a 1.2% change in voltage ,
so bias regulation sensitivity 1.2%/10% = 12% but accuracy is very high when bias current is lower due to lack of ESR drop and low variation in Vthreshold ( < 2%) same as a zener.
So I would choose 2~3mA
A typical 5.1V Zener is 60 to 80 Ohm 4.980 5.200 @ 5mA 80Ω max. but much cheaper in volume.
**broken link removed**