I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in \mathbb{N}$, where $\zeta_p(s) = (1-p^s)\zeta(s)$. Doing this, he defines the next function:
$ \zeta_{p,s_0}(s) = \frac{1}{\alpha^{-(s_0+(p-1)s)}-1} \int_{\mathbb{Z}_p^*}x^{s_0+(p-1)s-1}\mu_{1,\alpha},$ with $s_0 \in \{0,1,2,\ldots,p-2\}$ and $s \in \mathbb{Z}_p$. He then says that this $\zeta_{p,s_0}$ are 'branches' of $\zeta_p$. My question is: how does $\zeta_{p,s_0}$ define a branch ? $\zeta_{p,s_0}$ generally does not interpolate the values $\zeta_p(1-k)$ even for $k \equiv s_0 \bmod (p-1)$, since we have $k =s_0+k_0(p-1) \Rightarrow \zeta_p(1-k) = \zeta_{p,s_0}(k_0)$ and not $\zeta_p(1-k) = \zeta_{p,s_0}(1-k)$. So my question is really: what are the real branches of the $p$-adic zeta function ?