Polynomial for Dirichlet eta function

Clear[c, s]
(*c=10;*)
Exp[-s*Log[1]*c]
-Exp[-s*Log[2]*c]
Exp[-s*Log[3]*c]
-Exp[-s*Log[3]*c]


"Minimal analytic continuation part of Euler Maclaurin for Riemann zeta:"

(*start*)
k = 7;
s = 11;
1/k^(s - 1)
Exp[-(s - 1)*Log[k]]
%% - %
(*end*)

(*start*)
(*better version than below*)
Clear[s, c];
nnnn = 20;
Monitor[TableForm[Table[
   nn = nnn;
   Factor[Sum[
     k = kk;
     c = 1;
     a = Table[If[Mod[n, k] == 0, 1 - k, 1]*s/n, {n, 1, nn}];
     expk =
      MatrixExp[-c*
         Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n,
            1, nn}]][[All, 1]];
     (-1)^(kk + 1)*Factor[Total[Expand[expk]]], {kk, 1, nn}]], {nnn,
    nnnn, nnnn}]], nnn]
(*end*)


Clear[s, c];
TableForm[Table[
  nn = nnn;
  Factor[Sum[
     k = kk;
     c = 1;
     a = Table[If[Mod[n, k] == 0, 1 - k, 1]*s/n, {n, 1, nn}];
     expk =
      MatrixExp[-c*
         Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n,
            1, nn}]][[All, 1]];
     (-1)^(kk + 1)*Factor[Total[Expand[expk]]], {kk, 1,
      nn}]] nnn!, {nnn, 2, 12, 2}]]