Is there a table somewhere of the $n$th zero of $\zeta(s)$ for $n = 10^k$ for $k = 0,1,2,\ldots$? I need the values for $k$ up to as large as is known (e.g., $k = 22$). Same question for $n = 2^k$, or for other powers of a fixed integer. This is not found at http://www.dtc.umn.edu/~odlyzko/zeta_tables/, nor at https://www.lmfdb.org. Mathematica goes only to $10^8$. I need out to $10^{22}$.
EDIT: The context for this is that I'm doing some computations that require the zeros to some (not super-high) accuracy, and the values I'm asking for would be most useful. People have computed $\pi(n)$ for powers $n$ of a fixed integer $a>1$, so it shouldn't be too much to ask that we do the same for the $n$th zero of $\zeta(s)$.
These sequences can be computed at a guaranteed accuracy using ARB. This impressive C-library comprises of an example program to generate (ranges of) non trivial zeros $\rho$ of $\zeta(s)$ . The program uses a traditional method to compute zeros up till $10^{15}$ and then automatically switches to Platts's version of the Odlyzko–Schönhage algorithm. Below is what I generated on two desktop PCs in just a few hours, I'll leave it running to obtain some higher numbers and will add them in the next few days.
Added: updated the tables below with the desired higher values of $k$, i.e. $10^{25}$ and $2^{75} \approx 3.8 \times 10^{22}$. Also included a text file for $3^k..9^k$ that stops after $n^k > 10^{18}$ has been reached for each $n$.
$\,\,\,\,k \qquad \qquad \qquad \qquad \qquad \Im(\rho_{10^k})$
These last 5 values I had already computed a while ago.
$\,\,\,\,k \qquad \qquad \qquad \qquad \qquad \Im(\rho_{2^k})$