Its story is something like that the hero is is executed by the law for false reasons... might be for raping a couple in the forest... or like that and movie ends with his trgic execution... :(

Dead Man Walking... with Susan Sarandon and Sean Penn
(don't know that I would refer to him as a 'hero' though... he was guilty.. )

Maybe the Green Mile? With Tom Hanks?

Wow - There are a few movies with this plot. The only one I know of is THE GREEN MILE with Tom Hanks