Is the wavelength of the light emitted from an LED at its turn-on voltage greater than or less than the peak wavelength of the LED?
The wavelength distribution will look like the graph, pretty much regardless of current at a given temperature. LEDs are not purely monochromatic. 

However, as the die heats (and it will tend to heat more at higher current) the center of the spectrum will shift toward the red for all LEDs (longer wavelength). 

From this OSRAM document, you can see a typical change of about +70pm/K die temperature.