Although I had read Larsson's first Rebecka Martinsson novel, Sun Storm, ideally I probably should have read the two books that come between the first and the forth, as this one refers a lot to several people's backstory that occurs in one or the other of those. That said, though, you really don't need to have read the books in order to understand the crime story told in this volume.
Right from the outset we know that one of the main characters here has already died, but her strong narration of this event is one reason this book gets off to such an exciting start.
In terms of mystery, there don't end up being a whole lot of suspects, so if you're looking for a puzzle this may not satisfy you. The things I enjoyed about the story are different. First, Larsson writes of a certain mystical or spiritual element, and not just the immediate appearance of the dead girl. As another reviewer here writes, there is an element of redemption here, but it's not an easy fix and it certainly doesn't apply to everyone.
Second, Larsson apparently grew up in this small Swedish mining town of Kiruna herself and has a real gift for describing the natural beauty of its surrounds. And she is good on animals, particularly dogs, as well.
Finally, there is a lot of interesting Swedish World War II history that is part of the story, particularly Sweden's complicated relation to Germany, which it allied itself to for awhile. My impression is that this was not so much ideologically as in an effort to save itself from the Soviet Union, a nation they perceived as a bigger threat. There is a list of books she read to fill out this aspect at the back of the book, for those who want to know more.
(Originally posted on GoodReads.)