First and second zeta zero relation to tetration and Lambert

Clear[a, b, c, d, bigNumber]
a = RandomReal[] 10;
bigNumber = 10000000000000000000000000;
Monitor[Do[
  a = Round[(((2*Pi*Exp[1]/Im[ZetaZero[1]])^(2*Pi*
             Exp[1]/Im[ZetaZero[1]]))^-a)*bigNumber]/bigNumber;
  , {n, 1, 2000}], n]
N[a, 30]
b = N[-(a*Log[a]*Exp[1] - 1), 30]
N[2*Pi*Exp[1]*(1 - b)/Exp[1]/LambertW[(1 - b)/Exp[1]]*I, 30]

Clear[a, b, c, d, bigNumber]
a = RandomReal[] 10;
b = RandomReal[] 10;
c = RandomReal[] 10;
bigNumber = 10000000000000000000000000;
Monitor[Do[
  a = Round[(((2*2*Pi*Exp[1]/Im[ZetaZero[2]])^(2*Pi*
            Exp[1]/Im[ZetaZero[2]]))^-(a + b))*bigNumber]/bigNumber;
  b = Round[(((2*2*Pi*Exp[1]/Im[ZetaZero[2]])^(2*Pi*
            Exp[1]/Im[ZetaZero[2]]))^-(a + b))*bigNumber]/
    bigNumber;, {n, 1, 2000}], n]
N[a, 30]
N[b, 30]
b = N[-(2*a*Log[2*a]*Exp[1] - 2), 30]
N[2*Pi*Exp[1]*(2 - b)/Exp[1]/LambertW[(2 - b)/Exp[1]]*I, 30]

N[Table[(2*n* \[Pi] + Im[ZetaZero[n]] -
   Im[ZetaZero[n]] Log[Im[ZetaZero[n]]/(2 \[Pi])])/(
  2 \[Pi]), {n, 1, 12}], 30]