The ColorSpace for Lightroom Library and the Develop Module is different.
However I would suggest you to please check the monitor profile once of the computer.
Please have a look at the following article : http://www.lightroomqueen.com/articles-page/how-do-i-change-my-monitor-profile-to-check-wh ether-its-corrupted/
thanks - I checked it. However I have this problem for a few days know and never had it before. This is confusing me. Why does LR show me in develop mode a totally different picture in comparison to the library, although I didnt change anything.
The Library Module works on sRGB Colorspace and the Develop Module works on the ProPhoto ColorSpace.
Hm ok - and how I adapt it, so that the image looks the same?
This is most likely caused by a broken monitor profile.
Recalibrate your monitor, and the problem should go away.
You can, as a temporary fix, set the monitor profile to sRGB.
Is Windows 10 BTW?
The colorspace that the Develop module and the library module operate in is completely immaterial to what your image looks like. It doesn't matter and the difference is extremely subtle and only really visible on very high-end wide gamut monitors. Also, the colorspace in Develop is not prophotoRGB, it is a linear gamma space with prophotoRGB primaries called MelissaRGB which is quite different but again, not important for this.
We need to know one thing to further diagnose your issue: When you go back to Library, do the images stay dark or do they revert back to the lighter version? If so, you are dealing with several possibilities. The first is as already mentioned a bad monitor profile. You should recalibrate your monitor using calibration hardware. If you don't have that, test this possibility by assigning the sRGB profile to your monitor in the monitor's properties pane. If this fixes it, you need to get a calibration puck and calibrate your monitor, the issue is caused by a bad monitor profile that was installed by a system update. These are very common on the windows platform and even if other apps behave normal, this is the issue.
Second option is that your graphics card is not dealing with the GPU acceleration well and omitting color management. Test this by going into Lightroom preferences, selecting the performance tab and turning off graphics card acceleration. Restart Lightroom and check. (update: I see you're on LR 5.7: this option doesn't exist there so ignore this advise)
If neither of these fixed the issue and after you have gone to Develop, the image is also dark in Library, than several other issues might be happening. First is that you might have defined a camera default. Check whether any of the settings are non-zero on your images after import. They might be non-zero if you apply a preset upon import or when you set a default Develop setting. Reset the latter by going to Preferences->Presets and click "reset all default Develop settings". You're seeing the image change when you go into Develop because in Library, first the jpeg preview that the camera generated is shown and only when you go to Develop does the actual raw image get rendered by Lightroom.
Another issue that might be happening is that you use any of the dynamic lighting modes on your camera. Nikon calls these active D-lighting. You should turn those off completely. What the camera does when you have those enabled is to underexpose the image by a stop or so and then tag the raw image with a flag that tells Nikon software to increase the exposure compensation and to do a HDR treatment on the image to raise shadows and bring down highlights. Lightroom cannot read these tags and simply renders the image from the raw data (which is underexposed!). In Library you initially see the in-camera jpeg image and that image has been rendered by the camera using the HDR technique. You are better off exposing correctly and leaving these modes off if you want the highest quality raw data.