Tables of zeros of $\zeta(s)$

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 $\zeta(s)$.

Solution 1:

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 $n$.

$\,\,\,\,k \qquad \qquad \qquad \qquad \qquad \Im(\rho_{10^k})$

0   [14.134725141734693790457251983562470270784257115699243175685567460149963429809 +/- 3.93e-76]
1   [49.77383247767230218191678467856372405772317829967666210078195575043351161152 +/- 5.06e-75]
2   [236.52422966581620580247550795566297868952949521218912370091896098781915038429 +/- 4.55e-75]
3   [1419.4224809459956864659890380799168192321006010641660163046908146846086764176 +/- 3.25e-74]
4   [9877.782654005501142774099070690123577622468051781115996005448274058955511917 +/- 3.92e-73]
5   [74920.82749899418679384920094691834662022355521680155409349063157612661255158 +/- 4.38e-72]
6   [600269.6770124449555212339142704907439681912579061890094365456220213610910557 +/- 4.45e-71]
7   [4992381.014003178666018250839160093271238763581436814518246180779183990284991 +/- 1.17e-70]
8   [42653549.76095155390305030923281966798259513045217834410855322778683992559441 +/- 3.09e-69]
9   [371870203.83702805273405479598662519100082698522485040633971547149260604697270 +/- 9.19e-69]
10  [3293531632.3971367042089917031338769677069644102624896002918640087684198732839 +/- 8.61e-68]
11  [29538618431.613072810689561192671546108506486777642121547003676610028920767413 +/- 7.46e-67]
12  [267653395648.62594824214264940920070899588029633790156535642184489692722108891 +/- 5.00e-66]
13  [2445999556030.2468813938032396773514175248139258740941063719780888468673622153 +/- 5.20e-65]
14  [22514484222485.729124253904444090280880182979014905371993687787076797675433215 +/- 6.22e-64]
15  [208514052006405.46942460229754774510609948399247941058304066858487141797930129 +/- 7.30e-63]
16  [1941393531395154.7112809113883108073327538053720310680193595262570242168519486 +/- 6.78e-62]
17  [18159447720050928.218984697915762229855673976201939240381405179382305951827227 +/- 7.63e-61]
18  [170553583898990072.39238002790864220176542324287460556592405590364092212182419 +/- 6.15e-60]
19  [1607634529722392163.1447376508588766470866947935822065742294208556870244636240 +/- 2.52e-59]
20  [15202440115920747268.629029905502748693591388368618866827260757264806364151026 +/- 3.91e-58]
21  [144176897509546973538.2912188325359908282 +/- 4.6306e-88]
22  [1370919909931995308226.627511878006490223 +/- 2.2058e-87]
23  [13066434408793494969602.37457111931341385 +/- 1.6411e-79]
24  [124807082519145561455923.1051075922514060 +/- 1.1005e-36]
25  [1194479330178301585147871.329092007365198 +/- 4.986e-34]

These last 5 values I had already computed a while ago.

$\,\,\,\,k \qquad \qquad \qquad \qquad \qquad \Im(\rho_{2^k})$

0   [14.134725141734693790457251983562470270784257115699243175685567460149963429809 +/- 3.93e-76]
1   [21.022039638771554992628479593896902777334340524902781754629520403587598586069 +/- 6.03e-76]
2   [30.424876125859513210311897530584091320181560023715440180962146036993329389333 +/- 4.81e-76]
3   [43.327073280914999519496122165406805782645668371836871446878893685521088322305 +/- 7.15e-76]
4   [67.07981052949417371447882889652221677010714495174555887419666955169490121896 +/- 5.56e-75]
5   [105.44662305232609449367083241411180899728275392853513848056944711418149444758 +/- 5.59e-75]
6   [169.91197647941169896669984359582179228839443712534137301854144160780084147837 +/- 4.43e-75]
7   [283.21118573323386742049383794332893829845568487895863394524552244723946053379 +/- 6.22e-75]
8   [478.94218153463482653831710176703848402190424310960189553174422319680004296426 +/- 5.06e-75]
9   [826.9058109540807719796644509134941379920938463215759541736920400590372011978 +/- 4.33e-74]
10  [1447.2262270999757943922159349507653836806399104431413781107520698299942241563 +/- 3.42e-74]
11  [2565.9752704782838442040120469713348504654114815200091750889597502484566771312 +/- 4.16e-74]
12  [4597.820812342329887692851077511760921676080641718094770407772892584787490247 +/- 3.01e-73]
13  [8316.991697012572252064972124293399339922684951736532048263231166781013322366 +/- 3.11e-73]
14  [15162.369685178557186196366703441992579900552644223374188000832748816429426312 +/- 4.90e-73]
15  [27835.686158283640488122707304947696107697833511800659716198805513811913202623 +/- 3.97e-73]
16  [51408.764991841662119794205019781851376497853472604109912743136111735672796754 +/- 7.18e-73]
17  [95445.34909771818385815656281106820973255891996721271796216860855348500561548 +/- 3.46e-72]
18  [178028.79048273387593127749026393931184438703510584863654740342276359040847497 +/- 4.32e-72]
19  [333442.23294969070227365278780659904710246823460669947905617435364219133443458 +/- 5.77e-72]
20  [626834.9835264983966386596193718603971772950102553319215307436011302937121067 +/- 3.96e-71]
21  [1182295.1063641198903870943136432000864659652070439748448975686161939425793337 +/- 3.29e-71]
22  [2236650.2076711095250107458331814854148370365251323285340392768848517810218411 +/- 6.01e-71]
23  [4242761.036031418346077501762762736684538290734307777468880454492078976307870 +/- 4.25e-70]
24  [8068112.5426540081010234279535989083815760260652238675136484477033116273664401 +/- 8.49e-71]
25  [15377152.638247856238274226148193938623730694086678293478762300561834653778709 +/- 4.85e-70]
26  [29368339.096897685400157808579238150536685323766308244996296914176145141169718 +/- 7.28e-70]
27  [56196671.55493483696777384721551397296543001446591110730046706244048405325836 +/- 4.11e-69]
28  [107722370.35477129273932284583501189800244627656346545411000434541397083077454 +/- 3.27e-69]
29  [206827642.11037813303687956110731608326801511559427951607404106962536421559228 +/- 2.61e-69]
30  [397711241.25351041862407170289068366732153012975482565494613042121327725569712 +/- 6.90e-69]
31  [765840209.4465783056252302359688649854738113256664235220387410042068583090146 +/- 2.56e-68]
32  [1476652766.7292696395904363627266497692852171999806338638981158803819136197386 +/- 5.83e-68]
33  [2850699031.6481437701271385346513712925473139484271955739171674330483022871793 +/- 5.36e-68]
34  [5509642501.930849555381489634167680506856790217735034585204172555045660800109 +/- 4.06e-67]
35  [10660164024.237220934068897531970510812136808936683109354242036698535070110493 +/- 4.63e-67]
36  [20646422541.046792360756869221312635059625859955124342715901476735968145569531 +/- 7.05e-67]
37  [40025855798.87533340030159570256478987368301803633564062081911463984476672533 +/- 1.03e-66]
38  [77665455975.46881840143429735353428921580892190176868532935583538467441066302 +/- 5.85e-66]
39  [150829081672.87268087997227223267768482323573409106290604741955095637016213155 +/- 7.30e-66]
40  [293151666847.43529256407337775275897023080830612257570035637879270540848105469 +/- 5.72e-66]
41  [570205550406.9675818863385362424277408884768041075051124122232485586455128752 +/- 3.00e-65]
42  [1109903917486.5390623073785248231338047996220605280517351683594325307699293092 +/- 5.36e-65]
43  [2161914550339.1791435831650810474740346761492279157721003957602704753008817210 +/- 4.24e-65]
44  [4213823232653.1658479124517269308025179540134006827289363896471593064338734755 +/- 7.16e-65]
45  [8218361215543.2378945083553611019738623004078910168449064195328469387122475865 +/- 9.13e-65]
46  [16038090738210.389619761612230746936773105639144243299249744586503245009289454 +/- 6.22e-64]
47  [31316043489598.102327834599903298203026037687008886789864174336961671027439123 +/- 4.95e-64]
48  [61181048475701.80639196304655678171014427703137257832364884063320710470851384 +/- 2.04e-63]
49  [119589343968842.32518587882856435148973773539409437397028852588327619198310125 +/- 1.97e-63]
50  [233875093708608.87876160152862571697561001864757124334443721886372766115031012 +/- 3.11e-63]
51  [457596155084527.93030987221885020230317356998579944134633113583001471482566478 +/- 6.08e-63]
52  [895734261383610.3680074351759618167381589857223125885206258334374613658929009 +/- 2.46e-62]
53  [1754148932981883.9912741399373581214254158293922367223115750669875475427460566 +/- 5.01e-62]
54  [3436661011844225.5053936826736780914297128921954825607420810376661682932229376 +/- 4.55e-62]
55  [6735701151447302.958249660530007131555871581564196086543331227401017732099664 +/- 5.43e-61]
56  [13206816141597353.376177704540163514981431426220548421978487651964112827170849 +/- 6.36e-61]
57  [25904568090922928.493050534406816340161483749448375612861054863888125433298000 +/- 5.44e-61]
58  [50828994552669331.73558321975568822163909976234749986574114017763006847968752 +/- 1.95e-60]
59  [99769541839134379.54967406450632172766839579640371550043055834677669674594634 +/- 5.65e-60]
60  [195898172449255779.50521889994025543430258259980519085056917013294370758230645 +/- 6.12e-60]
61  [384772182854677902.63915264519255479311617699796394103152849388978594669575511 +/- 6.67e-60]
62  [755984709972501303.7006042486691771029804484523516301040911043653448485036603 +/- 2.82e-59]
63  [1485777731679926829.5755171624348977252192144523152979155073918556 +/- 8.78e-47]
64  [2920934497141236327.6040980320497162159057273242033845531381109545 +/- 8.16e-47]
65  [5743978322687433464.7587866399685869059920838792193895822299463886 +/- 6.06e-47]
66  [11298552994572559676.221241673796418915782606417578851012328613892 +/- 4.53e-46]
67  [22230445488615771848.757221959387856249813317405292513495940566099 +/- 4.71e-46]
68  [43750722198322584945.742573011403974192165870105147739241607752977 +/- 8.96e-46]
69  [86125268533375856544.378970276789855152484317048719846081827766251 +/- 5.14e-46]
70  [169582482199288958063.51192037756044955604550386711291268575630564 +/- 5.61e-45]
71  [333989874886948666896.88639808102772469292887987151863924763287235 +/- 4.62e-45]
72  [657937353290076703910.63965535573617484549380718934143891241209388 +/- 8.44e-45]
73  [1296378613004998735781.6214815290513946856812903451883241401015141 +/- 5.62e-44]
74  [2554891772189838945162.535569914980679845155488061266342479300386 +/- 8.16e-43]
75  [5036210474890574589384.9309614901772563830490135183728696357339310 +/- 8.66e-44]

Solution 2:

Not all zeros up to the $10^{22}$nd zero have been computed. The largest exhaustive zero computation that I'm aware of, where the first $N$ zeroes were computed, was done by David Platt in his paper on computing $\pi(x)$ rigorously, and is up to approximately the first $10^{12}$ zeros. The table of these zeros takes up approximately a terabyte and is stored on the LMFDB. I give the relevant data below.

Others have computed billions of zeros at semirandom but arbitrary heights. I believe this is where the number $10^{22}$ comes from, as Odlyzko computed several billion zeros of height around $10^{22}$ (though these are probably closer to the $10^{25}$th zero than the $10^{22}$nd.

I note that it is computationally infeasible to compute all zeros up to $10^{22}$, and storing them would take more than a trillion terabytes of space.

I take this data from the LMFDB database.