This could be a bunch of things. I doubt the fuel pump theory, but it is easy to check.

I would first suspect a flakey fuel pump relay. The car gets a whiff of gas from the cold start valve and fires, but the relay doesn't recognize that the engine is turning, and does not power the pump. Then you recycle the relay and the car starts. Easy to test - jumper pins 15 and 86 on the relay, start the car, and see if you still have the problem.

Next I would look at system pressure, which will tell you if you have a tired pump. Then control pressure at cold and warm start. There are very specific numbers for this depending on temperature.

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Chuck Taylor
Falls Church VA
'66 200, '66 230SL, '96 SL500. Sold: '81 380SL, '86 300E, '72 250C, '95 C220, 3 '84 280SL's '90 420SEL, '72 280SE, '73 280C, '78 280SE, '70 280SL, '77 450SL, '85 380SL, '87 560SL, '85 380SL, '72 350SL, '96 S500 Coupe