zeta zeros root form

(*start*)
Clear[number, log1, log2, log3, log4, s, x, X, h, n, k];
$MaxRootDegree = 2000
number = 100;
integer = 0;
m = 200;
s = Table[
   number*(2*Pi*I*integer -
      Log[Root[Sum[(-1)^(n + 1)*#1^Round[Log[n]*number], {n, 1, m}] &,
         k]]), {k, 1, Round[Log[m]*number]}];
Block[{$MaxExtraPrecision = 100},
 N[Sum[(-1)^(n + 1)*1/(E^(Round[Log[n]*number]/number))^s, {n, 1, m}]]]
N[Round[%, 10^-2]]
N[s]
Sort[Re[%]]
ListLinePlot[%]
(*end*)


(*start*)
Clear[number, log1, log2, log3, log4, s, x, X, h, n, k];
$MaxRootDegree = 2000
number = 100;
integer = 0;
m = 2000;
s = Table[
   number*(2*Pi*I*integer -
      Log[Root[Sum[(-1)^(n + 1)*#1^Round[Log[n]*number], {n, 1, m}] &,
         k]]), {k, 50, 50 + 5 + Round[Log[m]*number]*0}];
N[%, 12]
(*end*)

number*(2*Pi*I*integer -
   Log[Root[Sum[(-1)^(n + 1)*#1^Round[Log[n]*number], {n, 1, m}] &,
     k]])



(* later 9 8 2021 *)
(*start*)Clear[number, log1, log2, log3, log4, s, x, X, h, n, k];
number = 120;
integer = 0;
m = 100;
$MaxRootDegree = Round[Log[m]*number] + 10
$MaxExtraPrecision = Round[Log[m]*number] + 50
Sum[(-1)^(n + 1)*1/(E^(Log[n]))^s, {n, 1, m}]
s = ParallelTable[
   number*(If[k == 1, Pi*I, 0] + 2*Pi*I*integer -
      Log[If[k == 1, -1, 1]*
        Root[Sum[(-1)^(n + 1)*#1^Round[Log[n]*number], {n, 1, m}] &,
         k]]), {k, 1, Round[Log[m]*number]}];
Block[{$MaxExtraPrecision = 300},
 N[Sum[(-1)^(n + 1)*1/(E^(Round[Log[n]*number]/number))^s, {n, 1, m}],
   20]]
N[Round[%]]
N[s]
Sort[Re[%]]
ListLinePlot[%]
(*end*)

(* start *)

(* fungerande 27 12 2021 *)
Clear[number, log1, log2, log3, log4];
number = 12;
log1 = 0
log2 = Round[2/2*number]/number
log3 = Round[2/3*number]/number
log4 = Round[2/4*number]/number
gcd = GCD[log1, log2, log3, log4]
integer = 0;
s = 1/gcd*(2*Pi*I*integer -
    Log[Root[#1^(log1/gcd) + #1^(log2/gcd) + #1^(log3/gcd) + #1^(log4/
           gcd) &, 1]])
N[1 + 1/(E^(log2))^s + 1/(E^(log3))^s + 1/(E^(log4))^s]

(* end *)
"Output:"
-2.22045*10^-16 + 1.11022*10^-16 I


"Added 7 1 2023"

(*start*)c = 100;
x = N[Exp[-ZetaZero[1]/c], 100]
Sum[(-1)^k*x^(Log[k]*c), {k, 1, Infinity}]
(*end*)

"I managed to run the program with the c=2;
 " c " should be greater than Im[ZetaZero[1]]/Pi
 but already c=3 is too large to make to program
 run to the end."

(*start*)
Clear[h, x, k, s, s1, nn, m, c];
$MaxRootDegree = 2000;
c = 2
m = 10;(*m must be an even integer and greater than 4*)h = 200;
gcd = GCD @@ Table[Round[c*Log[n]*h]/h, {n, 1, m}]
Table[Round[c*Log[n]*h]/h, {n, 1, m}]/gcd
r = 100;
integer = 0;
s = 1/gcd*(2*Pi*I*integer -
    Log[Root[
      Sum[(-1)^(n + 1) #1^(Round[c*Log[n]*h]/h/gcd), {n, 1, m}] &, r]])
N[s, 80]
N[Sum[(-1)^(n + 1)/(E^(c*Round[Log[n]*h]/h))^s, {n, 1, m}]]
s1 = 1/gcd*(2*Pi*I*integer -
    Log[Root[
      polynomial =
       Sum[Sum[#1^k, {k, Round[c*Log[m - (2*q - 1)]*h]/h/gcd,
           Round[c*Log[m - (2*q - 2)]*h]/h/gcd - 1}], {q, 1, m/2}] &,
      r - 1]])
N[s1, 80]
N[Sum[(-1)^(n + 1)/(E^(Round[c*Log[n]*h]/h))^s1, {n, 1, m}]]
(*end*)

(*start*)
m = 20;
h = 200;
sort = Sort[
  Flatten[Table[
    Table[k, {k, Round[c*Log[m - (2*q - 1)]*h]/h/gcd,
      Round[c*Log[m - (2*q - 2)]*h]/h/gcd - 1}], {q, 1, m/2}]]]
"Plot of exponents of polynomial"
ListLinePlot[sort]
"Plot of coefficients of polynomial"
ListPlot[Sum[
  Table[If[sort[[n]] == k, 1, 0], {k, 1, Max[sort]}], {n, 1,
   Length[sort]}], Filling -> 0]
(*end*)

(*start*)
Plot[(1 + SquareWave[Exp[x]])/2, {x, 0, 2}, ExclusionsStyle -> Dotted]
(*end*)