a symbolic phrasing of the Riemann Hypothesis

Not to solve the million dollar problem, but just to look at the strongest way of phrasing it.

The Riemann Hypothesis says that the non-trivial (I'll define that in a bit) zeroes of the zeta function all lie on the same line, that line in the complex plane having real part 1/2.  the riemann zeta function is analytically extended in such a way (which I don't even understand yet) that the zeta of any even negative integer is zero.  zeta(-2)=0, zeta(-4)=0, zeta(-6)=0, and so forth.  To date, then, all points s of the complex plane for which zeta(s)=0  have either been the aforementioned trivial zeroes, or been complex numbers of the form a + bi, with a=1/2. they have 'real part' 1/2.  Let C denote the complex numbers, and Z denote the integers (never mind double struck lettering!)  Let C-Z be their set difference, and let є mean 'is an element of'.  So, a naive first attempt at phrasing the hypothesis itself might be:

1)   s  є  C - Z  =>  ( zeta(s) = 0  =>  Re(s) = 1/2)

Now to my question.  Could this symbolic phrasing of the question be strengthened, by changing conditionals to biconditionals?  For example:

2)   s  є  C - Z  <=>  ( zeta(s) = 0  =>  Re(s) = 1/2)     ,

3)   s  є  C - Z  =>  ( zeta(s) = 0  <=>  Re(s) = 1/2)     ,  or

4)   s  є  C - Z  <=>  ( zeta(s) = 0  <=>  Re(s) = 1/2)    .

Which of 1)-4) is the strongest, correct logical phrasing?  give counterexamples for the others.

2.

<=> in the parentheses in 3 and 4 is wrong since Re(s) = 1/2 doesn't imply that zeta(s) = 0; It would be sufficient to show one number c for which zeta(1/2 + ci) != 0 as a quick counterexample.

bump