Difficult cold start is usually tune up (or lack thereof). I would start with a bottle of redline, and then go from there. I dealt with the same issue for years with my m103 engine ( yours is m102 I believe) and ultimately it was solved with plugs, wires, cap rotor filters and most important new injectors.

It's an expensive nuisance to remedy. Good luck

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Eugene

10 E63 AMG
93 300te 4matic
07 BMW X3
14 Ford F-150 Fx2