live search issue

I am trying to implement live search in my site-currently when the user enters a search letter(for example)---the search results that appear are those that contain the letter regardless if it is found in the beginning,end or middle of the search result.

For example...if l is entered as a search term...Pistolou will appear in the search results despite the fact that this word starts from P and not from l.

I wanted to know if you think this is the correct strategy or maybe it is better the Pistolou appears only when the user enters the letter P as a searh term....like youtube does it or other sites out there.