Tables of zeros of $\zeta(s)$

Question

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 $\zeta(s)$.

Answer 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]

Answer 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.

$k$	$10^k$th zero of $\zeta(s)$
0	14.1347251417346937904572519835625
1	49.7738324776723021819167846785638
2	236.5242296658162058024755079556632
3	1419.4224809459956864659890380799166
4	9877.7826540055011427740990706901236
5	74920.8274989941867938492009469183467
6	600269.6770124449555212339142704907441
7	4992381.0140031786660182508391600932714
8	42653549.7609515539030503092328196679826
9	371870203.8370280527340547959866251910007
10	3293531632.3971367042089917031338769677068
11	29538618431.6130728106895611926715461085064

Answer 3

Mathematica contains the function ZetaZero[n]

I give you below the table for $n=2^k$ $$\left( \begin{array}{cc} k & \text{ZetaZero}[2^k]\\ 0 & 14.134725141734693790 \\ 1 & 21.022039638771554993 \\ 2 & 30.424876125859513210 \\ 3 & 43.327073280914999519 \\ 4 & 67.079810529494173714 \\ 5 & 105.44662305232609449 \\ 6 & 169.91197647941169897 \\ 7 & 283.21118573323386742 \\ 8 & 478.94218153463482654 \\ 9 & 826.90581095408077198 \\ 10 & 1447.2262270999757944 \\ 11 & 2565.9752704782838442 \\ 12 & 4597.8208123423298877 \\ 13 & 8316.9916970125722521 \\ 14 & 15162.369685178557186 \\ 15 & 27835.686158283640488 \\ 16 & 51408.764991841662120 \\ 17 & 95445.349097718183858 \\ 18 & 178028.79048273387593 \\ 19 & 333442.23294969070227 \\ 20 & 626834.98352649839664 \\ 21 & 1182295.1063641198904 \\ 22 & 2236650.2076711095250 \\ 23 & 4242761.0360314183461 \end{array} \right)$$

Above $k=23$, the argument is too large for the current implementation.

Answer 4

Existing calculations of the zeros of $\zeta(s)$ are summarized well on the Wikipedia page for the Riemann hypothesis (and citations to the literature are included). The first $1.2\times10^{13}$ or so zeros have been tabulated, and there have also been calculations of a relatively small number of zeros of very large height (around $10^{24}$, for instance, which is around the $8.3\times10^{24}$th zero). The Odlyzko–Schönhage algorithm is well suited for calculating specific zeros, so that might be where you can start to develop your own calculations if desired.

If you only want approximations to the zeros, the number of zeros of $\zeta(s)$ has an extremely good error term: for example, you could use this formula to find the $10^{22}$nd zero (which has size about $1.4\times10^{21}$) with an error of less than $10$; that's 20 significant figures, using an almost trivial calculation.