Clear[n, k, s, c];
n = 7;
s = N[14*I];
s - n*Limit[
1/Zeta[c]*
Sum[(-1)^(k - 1)*
Binomial[n - 1, k - 1]/Zeta[s + (k - 1)*(c - 1)], {k, 1, n}]/
Sum[(-1)^(k - 1)*
Binomial[n, k - 1]/Zeta[s + (k - 1)*(c - 1)], {k, 1, n + 1}],
c -> 1]

$s=14i$
$n=7$
$0.5 + 14.1347i = s-n \left(\lim_{c\to 1} \, \frac{\sum _{k=1}^{n} \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta ((c-1) (k-1)+s)}}{\zeta (c) \sum _{k=1}^{n+1} \frac{(-1)^{k-1} \binom{n}{k-1}}{\zeta ((c-1) (k-1)+s)}}\right)$
Ideally $n \rightarrow \infty$