My current **question** is concerned with a reference (paper or book) containing a proof of this result: The $k^{th}$ derivative of a L-function has necessarily infinitely many zeros.

This is a result of fairly standard complex analysis, probably due to Hadamard (entire functions of finite order strictly larger than one have necessarily infinitely many zeros). I don't have the book handy but this is certainly in Titchmarsh's "Theory of functions". It has very little to do with number theory.