Interestingly, according to the French AIP, L2K should only be contacted after Lille FIS, and failing that preferably LFAC or LFQQ. That does not always happen
RADIO CONTACT
Entry into French Airspace
...
The unit to be contacted
The radio contact must be established first with the flight information
centre (FIC) or flight information sector (FIS) concerned.
If it is impossible to establish radio contact with the FIC or the FIS
concerned when crossing the border, the pilot must contact preferably
the air traffic control units set up on the aerodromes, which are listed
below, in order to transmit them the «border crossed» message.
Border aerodromes concerned:
AJACCIO NAPOLEON BONAPARTE LFKJ
BALE MULHOUSE LFSB
BASTIA PORETTA LFKB
BIARRITZ-BAYONNE ANGLET LFBZ
CALAIS DUNKERQUE LFAC
CHAMBERY AIX-LES-BAINS LFLB
LILLE LESQUIN LFQQ
NICE COTE D’AZUR LFMN
PERPIGNAN RIVESALTES LFMP
STRASBOURG ENTZHEIM LFST