I will only add that the net installation took a looong time and there was no "Can't get line" error, the game started and only showed black screen. After copying the two files it runs well. Be patient, it tooks a moment to show the updater and the launcher. Good luck

Here's the guide:

1. Install and try to start the game: .NET Framework 3.5 error ("turn on or off")
2. Go to terminal and enter: protontricks 110800 --gui
3. From there, start winecfg (select default wineprefix) and set OS to Windows XP.
4. Close winecfg and open uninstaller: Uninstall Wine mono.
5. Exit protontricks GUI and try to start the game. .NET Framework 3.5 installer is working now. Wait for install to be finished, it will take some time. The game launcher starts and checks for updates. After that an error appears: "The Launcher cannot query DirectX. [...]"
6. Press OK, stop the game and all its processes, maybe even reboot to be sure. You can also do wineserver -k to kill the wineserver after correct wineprefix. (in my case WINEPREFIX=/home/sukulent/.steam/steam/steamapps/compatdata/110800/pfx)
7. Go to terminal and enter again: protontricks 110800 --gui
8. Open uninstaller and install DirectX from game files (game folder/3rd) provided.
9. Close uninstaller and open winecfg: Set OS to Windows 7.
10. Exit protontricks GUI and try to start the game - the launcher starts!
11. Choose whatever game settings you want, but set the renderer to DirectX 11.
12. You might get the "Can't get line" error or black screen. In that case go to: https://www.pcgamingwiki.com/wiki/L.A._Noire#Crash-to-desktop_on_Windows_8
13. Get the two files mentioned and copy them to the game folder (gamefolder/final/pc), overwrite if asked.
14. Try to start the game again