The function ζ has infinitely many zeros in the critical strip. This can be seen as follows.
Show that F(s) is an even function of s, and as a result, there exists G so that G(s2) = F(s).
(b) Show that the function (s − 1)ζ(s) is an entire function of growth order 1, that is
As a consequence G(s) is of growth order 1/2.
(c) Deduce from the above that ζ has infinitely many zeros in the critical strip.
[Hint: To prove (a) and (b) use the functional equation for ζ(s). For (c), use a result of Hadamard, which states that an entire function with fractional order has infinitely many zeros (Exercise 14 in Chapter 5).]