Well, the original Law & Order has gone through multiple cast changes. This site lists the various actors. Steven Hill, Jerry Orbach and Sam Waterston were each on the show for more than ten years, though only Waterston is still appearing. Here is an obituary of Jerry Orbach from a French website.
Law & Order is broadcast on French television under the name New York District. There are three spin-off shows:
[ul][li]Law & Order: Special Victims Unit (L&O: SVU). In France, known as New York Unité Spéciale (NYUS)[/li][li]Law & Order: Criminal Intent (L&O: CI). In France, known as New York Section Criminelle (NYSC)[/li][li]Law & Order : Trial by Jury (L&O: TbJ). I don’t think this has been broadcast in France, as it is still in its first season and has been (or soon will be) cancelled.[/ul][/li]
Among the four L&O shows, the charactors tend to move around pretty freely, as they all exist in the same “universe”. Since Fred Thompson plays the District Attorney for Manhattan, he could appear on any of the four programs.
All of the Law & Order shows take place in and around Manhattan. The typical structure of an Law & Order episode is that the first thirty minutes are spent with the police, investigating the crime, and the last thirty minutes are spent with the lawyers, conducting the trial. The other three shows vary, with L&O:SVU and L&O:CI concentrating almost exclusively on the police and L&O:TBJ concentrating almost exclusively on the trials.
This French website offers far more detail, including forums discussions. The OP is definitely not the only L&O fan in the republic.