There is a simpler way to derive the result from the approach mentioned by K. Conrad. You can find it in the following 1997 note by Paul T. Bateman:
Bateman, Paul T.: A theorem of Ingham implying that Dirichlet's L-functions have no zeros with real part one. 43 (1997), 281-284.
It is important to note that the consideration of the function H(s) can be traced back to a paper by Professor Narasimhan.