It sounds like your fan clutch is locked up. They'll be engaged when you first start the engine due to the fluid pooling, but should quickly unlock on a cold engine. Once the clutch unlocks, the fan speed won't ever rise much above it's idle speed regardless of engine speed until the air hitting the bimetal element in the center of the fan clutch reaches its operating temperature (meaning the radiator isn't keeping up with natural draft). Then, the clutch will engage, but they're designed to start slipping around 3000ish RPM, so it shouldn't sound like a "siren" at any point if it's working correctly.

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Current stable:
1995 E320 157K (Nancy)
1983 500SL 125K (SLoL)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)

Gone and wanting to forget:
1985 524TD 167K (TotalDumpster™) [Definitely NOT a Benz]