Some of the dead bodies are naked.
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Because The Following: Blood and Gore Nudity Sexual Themes Strong Language Use of Drugs Violence
No, because some nude scenes involve investigating dead bodies.
There are 21 missions in LA Noire but you can obtain/buy extra cases. There are also street crimes which are mini-cases, there are 40 street crime cases. The Complete Edition has all the pre-order and DLC cases, there are 26 in total, in full.
The complete edition includes several DLC missions, including The Consul's Car, A Slip of the Tongue (both now integrated into the main storyline), Nicholson's Electroplating and Reefer Madness. As well as this there are some control and performance improvements.
If I'm correct it takes place during the 1940s or 50s.