+1 ^ What Littlebettafish said.
I have mine in a dirt tank, so that's probably providing all the nutrient it needs right now. It's grown in clumps, too, as I don't really want it to spread fast, lol (though it's doing so anyway..). The dirt tank's too crowded with plants for a 'lawn' but I am thinking of replanting to the 3ft as one under some taller bog reeds.
Co2 is a carbon supplement, not the same as filter carbon. There's a whole science around that, not really my cup of tea as I am focussing on low-tech. Easy way is just what LBF said, dose with Flourish Excel from a bottle, which has liquid carbon in it.
My way of doing things is a bit slap-dash right now. Stick it in the tank, see if it grows..
In my smaller tank, which sounds pretty much like yours, Atena, I have LED lights too and only low-tech low light plants, java moss, java fern on mopani wood (get shrimps if you use mopani, it molds up fast, or use malaysian wood which doesn't as much), susswassertang.. a few small crypts and anubias, a little wisteria which does much better floating than planted. No room for hairgrass, but the little bit I grew in there was healthy, just not growing particularly quickly, not enough for a lush hi-tech lawn look, anyway.
So yeah, you may have to use Excel for that lawn.
Millifiore and Sowman, I'd be happy to chat about planted tanks over in my journal so's not to derial the OP's thread.. http://www.bettafish.com/showthread.php?t=92906&page=22
In which you can see a bit of hairgrass my wild bettas pulled up when flailing around in it, lol.