How are Zeta function values calculated from within the Critical Strip? We note that for $Re(s) > 1$ 
$$ \zeta(s) = \sum_{i=1}^{\infty}\frac{1}{i^s} $$
Furthermore
$$\zeta(s) = 2^s \pi^{s-1} \sin \left(\frac{\pi s}{2} \right) \Gamma(1-s) \zeta(1-s)$$
Allows us to define the zeta function for all values where
$$Re(s) < 0$$
By using the values where $$Re(s) > 1$$ But how do we define it over
$$ 0 \le Re(s) \le 1$$
Which is where most of the "action" regarding the function happens anyways...

An extension of the area of convergence can be obtained by rearranging the original series. The series

$$\zeta(s)=\frac{1}{s-1}\sum_{n=1}^\infty \left(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}\right)$$

converges for $\Re s > 0$. See here. 
