I found this (rather old) analysis from LinkedIn showing how people look at your profile on LInkedIn
Not surprisingly, most of the time was spent looking at your profile picture :)
However, continuing reading the article, I find they more or less conclude that whether to include a picture on your LinkedIn profile or not is a maybe!
I strongly disagree, but believe / hope that the reason is that this heat map and article is from 2012 and the way LinkedIn - and Internet - was used then is different from today
Today, of course you should include a profile picture on your LinkedIn profile!
Your picture is the first thing people see on your profile, and no picture will leave the impression of you that you don't take your career or professional network seriously enough - or they'll start to wonder why you don't want to show how you look...
If you - like 98 % of the rest of us - feel that you don't look on pictures, get the headshot done by a professional headshot photographer - then you for sure will look good :)