Why is my 3-prong dryer outlet showing 240V between hot and neutral/ground 'L' prong?
This answer was based on a prior version of the question

The picture that Jeremy posted in his answer you is perfect. I do however disagree with the idea that there may be a short between ground and neutral. Ground and neutral should be at the same potential and intentionally connected/ bonded in the breaker box.

As far as the readings you are getting L1 to Ground/neutral = 120vac and L2 to ground/neutral = 240vac, this leads me to believe there may be an interconnection of the two power leads. It would seem to me however, if they were somehow interconnected, you would see 240vac on either leg to ground. 

I would suggest taking the cover off the plug and testing the hot leads to the neural lug and case ground again. Obviously be careful and skilled with working with open wiring! The next step would be to take the same measurements at the breaker. It should be a double pole 30 amp breaker. Look for any double taps off one leg of the breaker. The readings should be 120 to ground/neutral from each side of the breaker and 240 across the hot lugs of the breakers.  If the same error exists, you may need to turn off the breaker, remove it and disconnect the wires and ohm out the individual paths looking for any shorts in the cable itself. This must be done with the wires disconnected from all power sources. Be sure to check for any voltage, even with wires off the breaker, before proceeding, just in case there is some other interconnection to a hot line.  

This trouble shooting is dangerous and should only be attempted by someone qualified. If you are not sure of your abilities, call a pro. 

I gotta think about this one a bit more. This situation and readings sound strange and improbable. Maybe one of my electrical guru buds on here can lend some better ideas.