I just want to say that having no memory of committing a crime isn't a "get out of jail free" card. It wouldn't even get you labeled as criminally insane and sent to an asylum.
When Light lost his memory, there was a time where L asked him something like, "Knowing what you know about this case, your own capabilities, the fact that at one point Light
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When Light got back his memory, even then he continued to play innocent so L just knew his behavior suddenly changed, but an observation like that is only an observation to keep an eye on and not probable cause for an arrest. L never trusted Light and was investigating him even when they were chained together, though the one thing L lacked was concrete evidence that Light and Misa had Kira's powers. The 13-day rule was supposed to knock the investigation off that track, though L never stopped suspecting him. However it was only a week of so between when Light got his memories back and Kira (Misa) started killing again. Light timed this on purpose: L has no concrete evidence and the 13-day-rule just trumps his theories further. L knew that Light was Kira, however he needed solid evidence to convict him. L did have a chance at getting solid evidence by testing the 13-day-rule and he openly announced this. As a result Rem wrote his name down to protect Misa, just like Light knew was going to happen.
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