According to the datasheet equation of the LM317, if R2 is 0, then Vout should always be 1.25V. The Wikipedia page of the LM713 says the same thing.
I have five LM317s, and if I ground the Adjustment pin, they all produce ~1.65 to 1.7V on Vout and not 1.25V. I tested with 1k, 10k and 100k load.
My test schematic is trivial: Vin = +5V (from a fancy power supply), ADj = GND, measure GND to Vout, plus .1uF and 1uF caps.
I'm looking for a temperature and Vin independent way to generate 1.25V reference and I thought since I had some of these they would work...
UPDATE: linear regulators need a load (see answer below). However the min current is 10mA, which is quite a bit for a battery op device. I found an LM385 which is a 1.2V diode shunt that only requires 10uA.