Let $\zeta(s)$ denote the non-continued, complex-valued zeta function defined on the complex plane where $\Re[s]>1$. Let $F(s)$ denote the continued zeta function defined on the entire complex plane (except possibly at $s=1$.) Is there not a way to write $F(s)$ explicitly in terms of $\zeta(s)$? My first inclination is to write something like...
$F(s) = \left\{\begin{array}{ll}\zeta(s) & \Re[s]>1 \\ 1 - \zeta(1 - s) & \Re[s] < 0 \\ ? & 0 \leq \Re[s]\leq 1 \end{array}\right.$
...but this may not be an accurate observation of the symmetry, if any, of the continued zeta function across the critical line. Furthermore, I do not know what to put for the question mark in the equation above, assuming anything could go there at all.
Is there anything reflective of the truth in what I've tried to write above? Is there an explicit formulation of the continued zeta function? How is the implicit formulation used in practice; say, to graph the continued zeta function?
The short answer is "yes".
For $\mathrm{Re} s < 0$, one uses the functional equation of the zeta function. This says that if we define $$ \Lambda(s) = s(1-s) \pi^{-s/2} \Gamma(s/2) \zeta(s),$$ then $$\Lambda(s) = \Lambda(1-s).$$ You can relate $\zeta(s)$ to $\zeta(1-s)$ (times a power of $\pi$ and a quotient of Gamma functions) in this way, and so the values of $\zeta(s)$ for $\mathrm{Re} s < 0$ are easily attainable from the values for $\mathrm{Re} s > 1$.
Within the critical strip, there are a variety of ways to try to understand the zeta function. You can check the answers to this related question or my answer to a related question for thoughts involving the eta function $\eta(s)$ (which is like an alternating zeta function) or the "approximate functional equation".