The largest zeta zero built into Mathematica 8 and apparently also in Wolfram Alpha as of 13.7.2021 is:

ZetaZero[10^7]
N[%]
0.5 + 4.99238*10^6 I

If one is happy with one significant decimal digit then this root function:

(*Mathematica start*)
Clear[f, s, n];
nn = 15;(*10^15 zeta zero*)f[x_] := Zeta[x];
(*The Franca-LeClair approximation of the zeta zeros:*)
n = 6;(*increase "n" for better precision.*)
(*The precision of N[11/8] needs to be increased too, accordingly.*)
s = 1/2 + 
   I*Table[2*Pi*Exp[1]*Exp[ProductLog[(10^n - N[11/8])/Exp[1]]], {n, 
      nn, nn}];
(*Root function for almost any function:*)
Monitor[
 z = Table[
   s[[j]] + 
    1/(1 - Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/
          f[k/n + s[[j]] - 1/n], {k, 1, n}]/
        Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/f[k/n + s[[j]]], {k,
           1, n}]), {j, 1, 1}], j]
Zeta[z]
(*end*)

can compute double that in order of magnitude ($10^{15}$), before the built in Riemann zeta function stops evaluating.

ZetaZero[10^15]
is approximately:
{0.580063 + 2.08514*10^14 I}

The formula is based on a conjectured formula where $s$ is the value from the Franca LeClair approximation for the $j$-th Riemann zeta zero made exact by the limit:

$$s+\lim_{n \rightarrow \infty}\left(\left(1-\frac{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s-\frac{1}{n}\right)}}{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s\right)}}\right)^{-1}\right)$$

The largest zeta zero built into Mathematica 8 and apparently also in Wolfram Alpha as of 13.7.2021 is:

ZetaZero[10^7]
N[%]
0.5 + 4.99238*10^6 I

If one is happy with one significant decimal digit then this root function:

(*Mathematica start*)
Clear[f, s, n];
nn = 15;(*10^15 zeta zero*)f[x_] := Zeta[x];
(*The Franca-LeClair approximation of the zeta zeros:*)
n = 6;(*increase "n" for better precision.*)
(*The precision of N[11/8] needs to be increased too, accordingly.*)
s = 1/2 + 
   I*Table[2*Pi*Exp[1]*Exp[ProductLog[(10^n - N[11/8])/Exp[1]]], {n, 
      nn, nn}];
(*Root function for almost any function:*)
Monitor[
 z = Table[
   s[[j]] + 
    1/(1 - Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/
          f[k/n + s[[j]] - 1/n], {k, 1, n}]/
        Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/f[k/n + s[[j]]], {k,
           1, n}]), {j, 1, 1}], j]
Zeta[z]
(*end*)

can compute double that in order of magnitude ($10^{15}$), before the built in Riemann zeta function stops evaluating.

ZetaZero[10^15]
is approximately:
{0.580063 + 2.08514*10^14 I}

The formula is based on a conjectured formula where $s$ is the value from the Franca LeClair approximation for the $j$-th Riemann zeta zero made exact by the limit:

$$s+\lim_{n \rightarrow \infty}\left(\left(1-\frac{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s-\frac{1}{n}\right)}}{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s\right)}}\right)^{-1}\right)$$

Mathematica code for the $10^7$ zeta zero: https://pastebin.com/tsJeGNs7

The largest zeta zero built into Mathematica 8 and apparently also in Wolfram Alpha as of 13.7.2021 is:

ZetaZero[10^7]
N[%]
0.5 + 4.99238*10^6 I

If one is happy with one significant decimal digit then this root function:

(*Mathematica start*)
Clear[f, s, n];
nn = 15;(*10^15 zeta zero*)f[x_] := Zeta[x];
(*The Franca-LeClair approximation of the zeta zeros:*)
s = 1/2 + 
   I*Table[2*Pi*Exp[1]*Exp[ProductLog[(10^n - N[11/8])/Exp[1]]], {n, 
      nn, nn}];
n = 6;(*increase "n" for better precision*)(*Root function for almost \
any function:*)Monitor[
 z = Table[
   s[[j]] + 
    1/(1 - Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/
          f[k/n + s[[j]] - 1/n], {k, 1, n}]/
        Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/f[k/n + s[[j]]], {k,
           1, n}]), {j, 1, 1}], j]
Zeta[z]
(*end*)

can compute double that in order of magnitude ($10^{15}$), before the built in Riemann zeta function stops evaluating.

ZetaZero[10^15]
is approximately:
{0.580063 + 2.08514*10^14 I}

The formula is based on a conjectured formula where $s$ is the value from the Franca LeClair approximation for the $j$-th Riemann zeta zero made exact by the limit:

$$s+\lim_{n \rightarrow \infty}\left(\left(1-\frac{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s-\frac{1}{n}\right)}}{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s\right)}}\right)^{-1}\right)$$

The largest zeta zero built into Mathematica 8 and apparently also in Wolfram Alpha as of 13.7.2021 is:

ZetaZero[10^7]
N[%]
0.5 + 4.99238*10^6 I

If one is happy with one significant decimal digit then this root function:

(*Mathematica start*)
Clear[f, s, n];
nn = 15;(*10^15 zeta zero*)f[x_] := Zeta[x];
(*The Franca-LeClair approximation of the zeta zeros:*)
n = 6;(*increase "n" for better precision.*)
(*The precision of N[11/8] needs to be increased too, accordingly.*)
s = 1/2 + 
   I*Table[2*Pi*Exp[1]*Exp[ProductLog[(10^n - N[11/8])/Exp[1]]], {n, 
      nn, nn}];
(*Root function for almost any function:*) 
Monitor[
 z = Table[
   s[[j]] + 
    1/(1 - Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/
          f[k/n + s[[j]] - 1/n], {k, 1, n}]/
        Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/f[k/n + s[[j]]], {k,
           1, n}]), {j, 1, 1}], j]
Zeta[z]
(*end*)

can compute double that in order of magnitude ($10^{15}$), before the built in Riemann zeta function stops evaluating.

ZetaZero[10^15]
is approximately:
{0.580063 + 2.08514*10^14 I}

The formula is based on a conjectured formula where $s$ is the value from the Franca LeClair approximation for the $j$-th Riemann zeta zero made exact by the limit:

$$s+\lim_{n \rightarrow \infty}\left(\left(1-\frac{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s-\frac{1}{n}\right)}}{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s\right)}}\right)^{-1}\right)$$