Since $\zeta$ has a single pole, at $z=1$, the radius of convergence of the Taylor series at $c$ is $r=|c-1|$. Moreover, if
$$\zeta(z)=\sum_0^\infty a_n(z-c)^n$$
is the Taylor expansion at $c$, then the limit in Hadamard's formula exists
$\lim|a_n|^{1/n}=1/r$ (this is an easy exercise: if a function has a single pole on its circle of convergence and no other singularities in a slightly bigger disk then the limit exists). Now a general <a href="http://www.math.purdue.edu/~eremenko/dvi/jentzsch.pdf">theorem of Jentzsch</a>
implies that the zeros of partial sums are:
a) those which tend to the zeros of $\zeta$ in this disc, and
b) additional zeros which are uniformly distributed near the circle $|z-c|=r$.

I don't think that this sheds any light on the zeros of $\zeta$.