Which is the best engineering course that I can do after completing my 12th with 56.8% marks? I have not given any B.Tech entrance exams.

Dear, You have not to be hopeless even if you did not take any entrance exam for engineering and with 57% marks. Don't worry, you can still do engineering, if you really wish to do it. You can opt for good private engineering colleges, but you must be very careful for choosing one as many are forge

Why don't you try for B.Sc in Computer Science? It is as good as B.Tech course and even from career perspective also it pays you well.