I am currently trying to use the LM317 with a stable output voltage of 1,25 V.
The general formula is
$$
Vo = Vref\cdot\left(1+\frac{R2}{R1}\right) + Iadj\cdot R2
$$

I used this circuit to create the 1,25 V output voltage:

simulate this circuit – Schematic created using CircuitLab
This circuit is directly according to the formula. R1 is infinitely high, R2 is zero. In general I measure the 1,25 V output voltage. But when the load is drawing current, everything gets unstable. Then I measure voltages higher and lower than 1,25 V...
I read some things regarding the minimum current in Adj Terminal. Therefore I thought about calculating R1 to create Iadj at ~50µA (according to ST LM317 datasheet). But whats with R2, should it be shortened to ground in this case?
I also thought about inserting a series resistor at the output, just to create a small load of ~1mA. Maybe the LM317 needs a minimum load... Are there any suggestions? Thanks for your help.