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Values of the logarithmic integral $\operatorname{li}(x)$
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Numbers
$x$ 
$\operatorname{li}(x)$
1/6:
-0.06547475339053838326644489563579973130355759446196068451605285903302198243939883141402067633084275766
1/5:
-0.08512648672879405370791083228202113835232859777962466211988925120354725425399056764513290329612445444
1/4:
-0.1186620564471231053050957064720401595749164603911061153126345432033962480473669408802575064828787141
1/3:
-0.1864113968315671433490242704890923488017397552111698672249113367472619189023740851810853601338468618
2/5:
-0.2529494192126212933022829920580060664558902027211945621653378184564285033783636660277323299681767647
1/2:
-0.3786710430610879767272071846365609805512340409782139969444209417345547567266746909858299806907975589
3/5:
-0.5468514142104169894910304389893463970509402509895959779963883847245338036334808762513598038738565521
2/3:
-0.6933084459253180677637371932728450081657115040342897044058565821841592580297134052773068301569920384
3/4:
-0.9369300110126454920969714373703949753450913066144732097911008211493125520106706612260564787111141941
4/5:
-1.134011957382327190850548677458555044081170454490742762857336106026243598827337348358509079736643961
5/6:
-1.299104475751430545084632992231803077786284411441559156100540327765580054059243837253821393199783062
7/6:
-1.132308231976449090881251650150200197737898223951259712799987998285790579178104455219791536912953589
6/5:
-0.9337872926672575108216844681723885603949407716466262586489399771235215547738387222122539185252951304
5/4:
-0.6864884538258715793229868045335851497842168436452206014779921012264938214203441170714581921689077543
4/3:
-0.3589138610719891223127107622769236175998320436324975503582038262334656521619344878567314811995344412
7/5:
-0.1449910047903065638201169133225021882717998829451998885644846027822883236657488616639341722238192775
3/2:
0.1250649863152963559943500047955129365420883239309922910956161087142330348062376541793459625335462869
8/5:
0.3537475506780349519235092384261775983121113047893365960228175736211267874576042237751104535235205181
5/3:
0.4897269891828197921518502740199958969537861833673137270452748760480711581059096338488782138591292418
7/4:
0.6454761842640260754159995042128083669552335086002962453994091154860710971291850944926347687868803260
9/5:
0.7326370311139213018432946759007848237209725696829595827685692074871885249573009765830508806714406283
11/6:
0.7884771583461672922873661437060096303278136621094181306450655800137811013470694612627198100278750118
2:
1.045163780117492784844588889194613136522615578151201575832909144075013205210359530172717405626383356
comment: This is the constant subtracted in Riemann's offset logarithmic integral $\operatorname{Li}(x)$.
13/6:
1.272548014062267259793772855283229397831645105428541953776965755939670193599905766344655305194764227
11/5:
1.315238277377661357620835384029446989937206406235256698435653000028642939582514203967901769734997544
9/4:
1.377763033714683621617664279151887722718877289388696515688686448257533259619517220990355991257040071
7/3:
1.478274681311237092017030366909289996691820062917629031148031859941913941865612656692018258026886231
12/5:
1.555670529102470395452913418398334049620089352404453872043876546943179091345026779884819483141462103
5/2:
1.667294667506323951359685970944537987257157048648050209868780913474143989751746107317171316270198995
13/5:
1.774144569343597950085088685880024808655614973022857804485563920805274645883894194463723494200788862
8/3:
1.843003100065812680994324705284003332186878486540779224712368047769351981867680192354749448809689996
11/4:
1.926653067030681748847318751443221788308652227559522722763016106305148609923004783269761847453141442
14/5:
1.975643342648437029540690044795191043655430682186830524084086561436931045737408777093205133530096492
17/6:
2.007832759190751437339705463964777277514515289112003964712680886703900558751087669367534610281467692
3:
2.163588594667191972876922367347721366542116212453868873985823001863092853305919390199866424856772226
19/6:
2.311648161353669067003491018253691854028550030977340860683005183288751276959539852455068175410407797
16/5:
2.340435501407820737503471970032934760020992857424293573382244399223880823609212126239509219731400067
13/4:
2.383137483803681407008827236161202592392221470941724375156709368282793590359637993861797239077539758
10/3:
2.453087757473175296649256365466270149393874853033814111428451201378569658050690070422630043140678185
17/5:
2.508008073741401186621598294643306166780953535434226600933589147087341602269463284910879712620570051
7/2:
2.588765078750509318095479562788679226278621665357891306912813130873245132898055046746052650748576193
18/5:
2.667700253587095438868102988371551311384626022336879039527386326173963820087970050128103546233697083
11/3:
2.719375244522756027965283000262590389966987809834301153898839594506051464099763590939067660521317340
15/4:
2.782962779738468478197013097526831291602348971887986250796646396224075189881449715709157044065241728
19/5:
2.820602553136195387180073616706659120121707370870734115405505220838020986189357857772390401033160868
23/6:
2.845489884752219295314088914411843655979717140827849582668757910521476317426228990274673953727103914
4:
2.967585095039050878010748878574128606562957836764878664147157425901540191961504623060409478832587968
25/6:
3.086061867178311567147361446564754494863562867809075328888200326874663961465963925814000870813423646
21/5:
3.109353940373963624474447993005651075502787616023641405739033333280748223196829457031175873845878228
17/4:
3.144051938431562297720284780196461256767223473081892702520416915315512231286294016592701869639397530
13/3:
3.201261307149793784694340730368683169956967108708698681073638062343690484631211273116586151523695408
22/5:
3.246490414767972530422308798956425917508971758442045036412458106304726752777901364500376473320797894
9/2:
3.313476163015205899591627932467062646567869177724049824771094287706301801675528352460177423564329400
23/5:
3.379479255033490872931323278452057336481003708234789677617631953517571985705470436341211896832149669
14/3:
3.422959689756067681617258420087773553514891466196076305753187841930821019725145548583972172547169774
19/4:
3.476747280020593716173554925360081698720988865381438388908109636957033184885119757686599619341535820
24/5:
3.508729194924042895602520762602697175206219802792510221762946597287856016529201025495611413330478632
29/6:
3.529932558873742471488722559028673020583923321408577693762151104161713195314255386425471614865033162
5:
3.634588310032651864392342706860811271370411550229815556665838870218151370805808813685376286938805576
Definition
For $x>0$ with $x\neq1$ the logarithmic integral is the principal value $\operatorname{li}(x)=\operatorname{PV}\!\int_0^x \mathrm{d}t/\log t$ [4]. This table holds it at rational $x$.
Parameters
$x$
—   argument ($x>0$, $x\ne1$)
Formulas
(1)
For $x>0$ and $x\ne1$, $\operatorname{li}(x)=\operatorname{PV}\!\int_0^x \mathrm{d}t/\log t=\operatorname{Ei}(\log x)$ [4].
Comments
(2)
Riemann's offset logarithmic integral satisfies $\operatorname{Li}(x)=\operatorname{li}(x)-\operatorname{li}(2)$ for $x>1$ [2].
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

RBF = RealBallField(numberdb.bits(100, losing=64))
RBF(QQ(2)).log_integral()
(P2)
Python
import mpmath

mpmath.mp.dps = 100
mpmath.li(2)
Links
Similar tables
Values of the exponential integral —   $\operatorname{li}(x)=\operatorname{Ei}(\log x)$ (1)
Wilbraham-Gibbs constant —   stores $\operatorname{Si}(\pi)$ and its normalisations
Stieltjes constants —   contains $\gamma_0$, the Euler-Mascheroni constant appearing in $\operatorname{Ei}$, $\operatorname{Ci}$ and $\operatorname{Chi}$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds $\operatorname{li}(x)$ at every rational $x=a/b$ in lowest terms with $b\leq6$ and $0<x\leq5$, except the singular $x=1$)
How they were obtained:

Each value was computed as a Sage real ball with arb at numberdb.bits(digits, losing=64) bits, and checked against mpmath at 150 decimal digits and against the OEIS value for $\operatorname{li}(2)$ [7]. The widest returned ball had radius less than $10^{-117}$ for 100 requested digits.



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