This. In fact, I am currently in the process and I finally chose to have a generic one with a LaTeX to adapt to company names and recruiters and I wrote special

This. In fact, I am currently in the process and I finally chose to have a generic one with a LaTeX to adapt to company names and recruiters and I wrote specialized when I can see who is going to read my cover letter. The third option I use is when I contact the recruiter by mail, I provide the generic one as attachment and in the mail provided more specific motivation and experience related to the job offer. And most of the time now, companies don't want you to upload your cover letter but use a form with a textarea box the size of a stamp to write your cover letter without any formatting option.