Characteristic Polynomial alternating series Riemann

(*start*)
number = 10;
log1 = 0;
log2 = Round[Log[2]*number]/number;
log3 = Round[Log[3]*number]/number;
log4 = Round[Log[4]*number]/number;
Clear[s];
x = s /. FindRoot[
   1 - 1/(E^(log2))^s + 1/(E^(log3))^s - 1/(E^(log4))^s == 0, {s,
    1 - I*20}, WorkingPrecision -> 100]
1 - 1/(E^(Log[2]))^s + 1/(E^(Log[3]))^s - 1/(E^(Log[4]))^s
s = x;
"(1) alternating series"
1 - 1/(E^(Log[2]))^s + 1/(E^(Log[3]))^s - 1/(E^(Log[4]))^s
1 - 1/(E^(log2))^s + 1/(E^(log3))^s - 1/(E^(log4))^s
Clear[x]
1 - x^Round[Log[2]*number] + x^Round[Log[3]*number] -
 x^Round[Log[4]*number]
"(2) alternating polynomial"
X = Exp[-s/number]
1 - X^Round[Log[2]*number] + X^Round[Log[3]*number] -
 X^Round[Log[4]*number]
(*end*)
h = {log1, log2, log3, log4}*number
"(3) Characteristic polynomial"
Expand[CharacteristicPolynomial[
    Table[Table[(-I)^(n + k - 0)*x^((h[[n]] + 1)/2)*
       x^((h[[k]] + 1)/2), {k, 1, 4}], {n, 1, 4}], x]/x^4 - 1]
(*end*)