# Why doesn't the Riemann Zeta Function have a zero at $s=0$?

I have a few questions regarding the Riemann Zeta Function and its zeros. First, I will state what is clear to me. I see that $$\zeta (s)$$ is clearly defined and nonzero when $$Re(s)>1$$ since then $$\zeta (s)$$ is defined by convergent power series. I know that $$\zeta (s)$$ has a pole at $$s=1$$ of residue $$1$$. Likewise, I am familiar with the function equation $$\pi ^{-\frac{s}{2}}\Gamma (\frac{s}{2})\zeta(s)=\pi ^{-\frac{1-s}{2}}\Gamma (\frac{1-s}{2})\zeta (1-s)$$. From this equation, I see that $$-2,-4,\dots$$ are zeros, since $$\Gamma$$ has poles there. I also know that we refer to the the nontrivial zeros as those in the critical strip, i.e. $$0. Here are my questions:

1. How do we know that there are no nontrivial zeros to the left of the critical strip, i.e. when $$Re(s)<0$$?
2. How do we know that there are no zeros where $$Re(s)=0$$?
3. Why is $$s=0$$ not a zero of $$\zeta (s)$$? This question is particularly key to me, since $$\Gamma(s)$$ is not defined at any nonpositive integer. By the same reasoning we used to deduce that the negative even integers are zeros using the functional equation above, shouldn't $$0$$ also be a zero? Since $$\Gamma(s)$$ is not defined, I don't see how this doesn't violate the functional equation.
4. How do we know that $$\zeta (s)$$ has no zeros on the line $$Re(s)=1$$?

I understand that I am literally asking multiple questions, but I am hoping that a single clarification will help me answer all four.

1. If you know that $$\Gamma(s)$$ doesn't have zeros (i.e. $$1/\Gamma(s)$$ is an entire function) and that $$\zeta(s)$$ doesn't have zeros in $$\Re s>1$$ (because it is represented by Euler product there), then the answer follows from the functional equation: $$\Gamma(s/2)\zeta(s)$$ doesn't have zeros in $$\Re s<0$$.
3. Again, follows from the functional equation (if $$s=0$$ were a zero of $$\zeta(s)$$ then $$\Gamma(s/2)\zeta(s)$$ would stay regular at $$s=0$$).
1. Yes, $$\Gamma(\frac{s}{2})$$ has a pole at $$s=0$$. However, calculating $$\zeta(s)$$ at $$s=0$$ cannot be settled with that argument alone due to the fact that $$\zeta(1-s)$$ also has a pole at $$s=0$$, meaning that you'd need to resolve an $$\frac{\infty}{\infty}$$ situation to calculate the actual value of $$\zeta(s)$$.
1. Consider the functional equation you give, along with some general knowledge of the $$\Gamma$$ function (it has no roots, and only poles at non-positive integers). No roots for $$Re(s)>1$$ then implies no roots for $$Re(s)<0$$, apart from the trivial ones.