Does the story of LoS integrates with the rest of the franchise?

I'm a huge fan of the Castlevania franchise. I played nearly all of the games, my favorites being Aria of Sorrow and Symphony of the Night. I didn't want to play the Lord of Shadows game serie because of its gameplay and "level design". Unlike Order of Ecclesia where you have several levels but need to come back to some of them with new abilities, LoS concentrates on the "one shot level" system (I'm inventing a lot of terms right now I think).

I started by playing the 3DS game (LoS Mirror of Fate) and found the story truly captivating, we learn the origins of Alucard, and lots of other things. Next, I played the first LoS game, the story is amazing too, but now I wonder, does the story of LoS integrates with the other games or is it a totally different story ? I did a bit of research but it seems I can't google correctly.

Could anyone tell me if the story of LoS (the whole game serie, including LoS, LoS:MF, LoS2) matches the story of the other Castlevania games ? (Especially with the Belmont chronology or things like that)

I didn't play LoS2 so I'd like to avoid spoilers about that particular game. Thanks !