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:

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