interesting, those are the exact symptoms on my car except mine will stay on for long periods of time after a change in weather. it's been on solid these past few days. i attribute it to a bad connector or faulty temp sensor unit behind the instrument cluster.
assuming it is a similar design to w124 - you can do a simple test by disconnecting the probe lead at the junction inside the fuse box and placing a 3k ohm load across the terminals leading back inside the car. if the display comes on then it's the probe that's faulty, otherwise you have to check further upstream which is basically the display unit behind the cluster.
i've gone so far as to replace the thermistor which didn't fix my problem. a new unit is about $200 - i'm not really that eager to replace mine yet...
1993 300e-2.8 - gone now <sigh>
"Do not adjust your mind, it's reality that's malfunctioning"