Basically, after the 2nd movie ended, we see L writing his name down in the Death Note in order to stop Kira (Yagami Light). He wrote that L Lawliet, will die in 23 days of a heart attack peacefully.
However, even if he's about to die and all, L cannot have a peaceful time. Instead, he's faced with another case. However, while the Kira case was a serious battle of wits and intellect, the case for this time is seriously quite degrading for our dear super sleuth.
Watari dies within the first 10 minutes of the show, presumably the work of one of the shinigamis. We also get to see a cameo of Misa Amane, which I think she gets off pretty much scot-free as she's a popular celebrity and what judge would believe that she goes around killing people through supernatural means?
So, L is all alone now and going to die soon. What does he do? The usual routine in his life, like duh. Consuming enough sugar to kill someone at an alarming rate, he makes use of his time wisely by solving various crimes that have stumped the Interpol and all. Anyway, that's not the point! Fans of L already know the routine of L's life!
Hence, moving on, we see a village in Thailand struck by disease. They're all dying and bleeding from various orifices and broken skin and all. Very gross anyway. Anyway, some military forces then move in to destroy the plague-ridden village. However, we see that there is an agent of the Wammy House, F who is also involved in this yucky situation. He discovers a strange boy who is apparently unaffected by the disease at all and they escape the village where F gives the boy some instructions and later dies because he's diseased already.
Anyway, said boy follows the instructions carefully and all, and is then sent to L for the child was a gift to Watari. And since Watari is dead... the gift now falls to L.
I'm a little lazy to summarize every single thing so this summary shall be more concise.
The next person to show up on L's doorstep is 12 years old Nikaido Maki who is the daughter of Professor Nikaido, an expert on viruses and had actually made a vaccine. Though he killed himself in the end to protect the vaccine from the terrorist group. With her is a strain of the same virus that wiped out the Thai village. And behind her are a terrorist group that desires the virus so that they can eliminate the human race to create a better world. Oh and the fact that they'll make a sweet and tidy sum at the end, it's not the main goal, seriously. They just want to change the world for the better. Really.
The theme of changing the world and creating a better place is similar to Light's goal. However, this theme isn't exactly fully explored. And also, Light and L are considered geniuses, while the antagonists in this movie aren't really that smart... so the gripping battle of wits where Light and L used to outsmart each other are missing. Which is really a very wasted.
Okay back to summary because who really wants to hear me ramble on. The plot is basically very cliche. You got a boy who apparently cannot be affected by the virus. You got a girl who has a vial of the virus. You got a terrorist group out for the virus. What next? Please don't let me underestimate your intelligence.
SPOILERS, WHAT SPOILERS?
Anyway, the main goal of watching this movie is to see L. That's all for me. However, while the movie does explore character development of L, I... feel that at certain points, L's pretty out of character. But hey, I get to overdose on the quirkiness and cuteness that L is. And the fact that we actually get to see L cycling and running and... trying to straighten his back... it makes my inner fangirl pretty much satisfied.
Just don't expect too much from this movie. There is no opponent for L to outthink or outwit because he's already smarter than the lame terrorist group, unlike Light who is a much more worthier opponent.
And what makes me glad is the way that they pay tribute to the manga by naming the Thai boy Near and giving him the position of L's successor. And yup, I sort of guessed it halfway through the movie, once the boy started to play with the robot and the sugar cubes. I mean, that's so very a Near trait...