Why are these numbers close to $-\log(2)+\text{integer}\,i\pi$?

Question

The following function $f(n)$ has been derived from the Dirichlet eta function:

$$f(n)=\log \left(\sum _{k=1}^n (-1)^{k+1} x^{c \log (k)}\right)-c \log (n) \log (x) \tag{$\ast$}$$

Let: $$s=\rho _1$$ be the first Riemann zeta zero,

and let: $$c\geq \frac{\Im(s)}{\pi }$$ and let: $$x=\exp \left(-\frac{s}{c}\right)$$

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For $$n=10^j: \; \Re(\exp(f(n))) + i \frac{\Im(f(n))}{pi}$$ I get the outputs

$n=10^0: 1 + i 0$
$n=10^1: 0.6181958706 + i 10.7638191193$
$n=10^2: 0.4999855568 + i 20.9774479431$
$n=10^3: 0.4998874715 + i 30.9977498270$
$n=10^4: 0.4999876248 + i 40.9997750332$
$n=10^5: 0.4999987512 + i 50.9999775038$
$n=10^6: 0.4999998750 + i 62.9999977504$

To me there is a trend in the decimal digits towards real part $\Re{} = 1/2$ and imaginary part $\Im{} = \text{integer}\,i$

Question:

Does the pattern/trend: $\lim\limits_{n \rightarrow \infty} \log \left(\sum _{k=1}^n (-1)^{k+1} x^{c \log (k)}\right)-c \log (n) \log (x) = -\log(2)+\text{integer}\,i\pi$ continue only for $s=$ a Riemann zeta zero?

(* Mathematica start*)
Clear[x, n, k, c, kk]
n1 = 10^4;
n2 = 10^4;
z = 1;
s = ZetaZero[z];
c = Im[s]/Pi; (* c >= Im[s]/Pi *)
c = 100;
x = N[Exp[-s/c], 20];
Monitor[table = 
   Table[Log[Sum[(-1)^(k + 1)*x^(Log[k]*c), {k, 1, n}]] - 
     Log[n]*Log[x]*c, {n, n1, n2}];, k]
Exp[Re[table]]
Im[table]/Pi
(* end *)