Values of the logarithmic integral $\operatorname{li}(x)$

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Numbers

$x$

$\operatorname{li}(x)$

1.045163780117492784844588889194613136522615578151201575832909144075013205210359530172717405626383356

comment: This is the constant subtracted in Riemann's offset logarithmic integral $\operatorname{Li}(x)$.

10:

6.165599504787297937522981752669522749130602806376581578932338490205589934760180157228276724309143254

100:

30.12614158407962992590174133903218497959990703890315154216356044099095339144004461735322234139553627

1000:

177.6096579901522266876406239486993179785577025645480810681284511852751362573542630364773157199519343

10000:

1246.137215899388459692771107529059792486534653514072285774809081409016981733687861836397912132822135

100000:

9629.809001050798205034259560580094958897668987629031163363379794260332987106089243105737290935056553

1000000:

78627.54915946218191986291074794726116132187438242176707475931789394167624357802172453432702851546268

10000000:

664918.4050485689123292128866011957931533265951242896704016376596538895257268980267843693643510260594

100000000:

5762209.375448031467569073609368981565599778125345746898577936231379153622019481878609558922124405586

1000000000:

50849234.95700179800400878908022360785563359242887901959079424663250024819164245918838010382482384246

10000000000:

455055614.5866230756095294565839217624488629330581957006107933942661284107341951220768250897123004853

100000000000:

4118066400.621611515039435328363888982149828490104138254840340007343271232824015426400621634421194178

1000000000000:

37607950280.80486548953492709097818869279517315682222599982647854380532130506337047267215579865356049

10000000000000:

346065645810.0503352144904675396792666603457677866237607904466014788649036628244926752371229782648783

100000000000000:

3204942065691.953988072328915392097804375703608868849568620990123082154144406649958448052111992287977

1000000000000000:

29844571475287.58106490589058598594308149535013999906466925702605488461494751932061564151921187834778

10000000000000000:

279238344248556.7927813936332216188213887549024553079095023638798116214373662550455374357157306057292

100000000000000000:

2623557165610821.778069124873574285405756091484733160998097090634270440095820712006083867586053646908

1000000000000000000:

24739954309690415.02218938383725843483282112195906457058271503009713781540303193599378308818585604821

10000000000000000000:

234057667376222382.2232510309771484401121240823077407408753245166317451273058727287787502679313142346

100000000000000000000:

2220819602783663483.548305532067778823947751781788595977681676146360558159071139757660984123841602350

1000000000000000000000:

21127269486616126182.33305833593592352819698972996559598656203540556796488788914583864214152788382314

10000000000000000000000:

201467286691248261498.1505286717817039976105789431935817938808923600260738959111839625517552880452760

2.163588594667191972876922367347721366542116212453868873985823001863092853305919390199866424856772226

3.634588310032651864392342706860811271370411550229815556665838870218151370805808813685376286938805576

4.757051765997606069292390575450131824861568757165315950696408010698927727109200529964652542349845415

11:

6.591009215721214087782454444587523253210901960225267317131158051648235128544200027474424576960244917

13:

7.396547948188126599235034804933501798495305794791972618374405168216896334745751219251436437199606983

17:

8.876464600018383278257414602263823491591318320619044136398149327861307200465228798123964702539690378

19:

9.568625832887769196695123543746984835891510919840938530378155368833341761846179513199042868305026966

23:

10.88354705428284581621873343958236761167006069765757525357048878522907689079539630871534751961141106

$e$:

1.895117816355936755466520934331634269017060581732707591646228431882513834533804153548900710126138957

$e^{2}$:

4.954234356001890163379505130227035275518053562420042054527095254570707913298407190883096971852850163

$e^{3}$:

9.933832570625416558008336019216765262990653022987582119693034065242496228332697443187187398902087851

$e^{4}$:

19.63087447005622002264572027972383885313207975115315216815404620859056206761114761767084729515828299

$e^{-1}$:

-0.2193839343955202736771637754601216490310472934069082075779786130735686985591415447222102510351372500

$e^{-2}$:

-0.04890051070806111956723983522804952231449218496302311632732287371169292871415219127926896100745164177

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 selected real $x$.

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].

(3)

The existing decimal powers remain the explicit prime-counting benchmarks of [8], not a new decimal mesh. The first nine primes, including the retained argument $2$, provide a bounded reference at the first jumps of prime counting; this is our exploratory selection, not a claim that the source tabulates those inputs. The exact substitution $\operatorname{li}(e^m)=\operatorname{Ei}(m)$ selects short exponential arguments on both sides of $1$ and distinguishes the unoffset convention [4]. The exponent list is a finite reference choice, not a completeness claim.

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

[1]

Wikipedia: Exponential integral

[2]

Wikipedia: Logarithmic integral function

[3]

Wikipedia: Trigonometric integral

[4]

DLMF 6.2: Definitions and Basic Properties

[5]

DLMF 6.4: Analytic Continuation

[6]

DLMF 8.19: Generalized Exponential Integral

[7]

OEIS A069284: decimal expansion of $\operatorname{li}(2)$

[8]

Oliveira e Silva, Computing pi(x): the combinatorial method (2006), Table IV

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 preserves $\operatorname{li}(2)$ and the published $10^n$ benchmarks for $1\leq n\leq22$, and adds primes $3\leq p\leq23$ and $e^m$ for $m=-2,-1,1,2,3,4$ as exploratory structural references on both sides of the singularity at $1$)

How they were obtained:

All retained numeric strings, exact values and row annotations are unchanged. The standalone generate.py, balanced_values.py and selection.json reproduce the entire current selection. Each finite Arb enclosure at 733 bits must fit inside the interval claimed by the written last place; a second run uses 912 bits. New nonexact entries have 100 significant digits.

Every new entry has an independent mpmath quadrature, closed-form or inverse-function check at 150 and 200 decimal digits, with a nonzero known-integral control. Arguments are exact rationals or propagated balls from exact expressions; root arguments use strict sign-certified rational brackets. Lambert and saddle roots also pass endpoint sign tests on the written last-place interval. The measurements are in coverage-checks.json. Older attachments are retained for provenance; their previous enumerations are historical, and the active enumeration is selection.json.