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Values of the polylogarithm $\mathrm{Li}_s$ at rational arguments
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Numbers
$s$
$x$ 
2
1:
1.644934066848226436472415166646025189218949901206798437735558229370007470403200873833628900619758705
comment: $\zeta(2)$.
2
-1:
-0.8224670334241132182362075833230125946094749506033992188677791146850037352016004369168144503098793527
comment: $-\zeta(2)/2=-\pi^2/12$.
2
1/2:
0.5822405264650125059026563201596801087441984748061264254343470478731710440716832008168403185879158572
comment: $\pi^2/12-\log^2(2)/2$.
2
1/4:
0.2676526390827326069191838284878115758198570669138545938652013531126933436319257768539503068025250930
2
-1/2:
-0.4484142069236462024430644059157743208342699413491991285017463713168243722557203123898651651866533107
comment: The negation $-\mathrm{Li}_2(-1/2)$ is OEIS A355234 [5].
2
1/3:
0.3662132299770634876167462976642627638020634155896782205109153133448785067521492485652493476090744827
2
1/9:
0.1143602069785100305994265765428921179929440377536944324111022613874716145144789981827825861863388190
2
2/3:
0.8332718864773899574410124619689003974407247623402611102588805048359080318734051703144657162693642234
2
3/4:
0.9784693929303061037430666665245614977614842746194872521048291995006417637926148071891464701017043170
2
-1/3:
-0.3090331264878084723170330093928167048055913967128310043053641826511426994949097494738580545159050732
2
-1/4:
-0.2359002976862634538210827939490320839289110881090912703320605109993961996295972637890600116444286107
2
-2:
-1.436746366883680946362902023893583354249956435654872102667243924865015789277397797543737867155068890
2
-3:
-1.939375420766708953077271719177891441222590177808578425838557466747972528312283751589042996989558299
2
0:
0
3
1:
1.202056903159594285399738161511449990764986292340498881792271555341838205786313090186455873609335258
comment: $\zeta(3)$.
3
-1:
-0.9015426773696957140498036211335874930737397192553741613442036665063786543397348176398419052070014436
comment: $-3\zeta(3)/4$.
3
1/2:
0.5372131936080402009406232255949658266704024993403781706897619307183240809201383973304123599754396700
comment: $7\zeta(3)/8-\pi^2\log(2)/12+\log^3(2)/6$.
3
-1/2:
-0.4725978446588968746186231931265476736649961609583811588208017271806097560412283668836298685192109158
3
1/3:
0.3488278611548400842142519353869928553662453745265499098803075362880099466100770380711690370205803195
3
2/3:
0.7380606448308579106637761463656503254736730698535132985805452330704338094395960663582315718000510060
3
1/4:
0.2584613957965733052880001298736726120216253535279880474758408141508572995156401217871299658249150170
3
3/4:
0.8444258088622044485043448773485128867039564982353540997492742750145177805559933697174281689418254328
3
-1/3:
-0.3206509480051539513225254693626089175914981464315212043063505123102905035381688787105406347897662253
3
-1/4:
-0.2427120033389163051507508979053807573340934214587924412587913956610225071574082049869655156245879566
3
-2:
-1.668283363966571212046345315887518776240980416547801186773359009104104716629034991977380865557780580
3
-3:
-2.348790554584076557805870669806798778113724842812620411209755519875395046369344664668805039878816614
4
1:
1.082323233711138191516003696541167902774750951918726907682976215444120616186968846556909635941699917
comment: $\zeta(4)$.
4
-1:
-0.9470328294972459175765032344735219149279070829288860442226041885136055391635977407372959314489874276
comment: $-7\zeta(4)/8=-7\pi^4/720$.
4
1/2:
0.5174790616738993863307581618988629456223774751413792582443193479770082815818649769364857778265609006
4
-1/2:
-0.4857145378306064445732989477546728727548964970716950127595560700348977663341528655864584738048938885
4
1/3:
0.3407911308562507524776409440122023152122808582868996278686462741215169180720208608409444536189947886
4
2/3:
0.6991935944929980954070226084971812195153493475154529099729317668369449609696398133480708820857111249
4
1/4:
0.2541161907463435340596737131535205829398478245574739638781062235368841219816968908002184321733360974
4
3/4:
0.7922210279728277795294857895573574111673948085877831608728044783196876015638504746179389262955397478
4
-1/3:
-0.3268035964621760559402001146190280627989065275875900550451982734844996203038913908257253642745586323
4
-1/4:
-0.2462727887607290643683494888096253547473037638415183658524785763802653421722296204125694622210012179
4
-2:
-1.813126015328491172128145260606351874682869166503313427982577253024631854666351230692237462446589213
4
-3:
-2.620634701909763503399941391847464354791954950264060636806216802516638146229723429178897603147980355
5
1:
1.036927755143369926331365486457034168057080919501912811974192677903803589786281484560043106557133336
5
-1:
-0.9721197704469093059356551435534695325535133620330432612258056355348158654246388917750404123973125029
5
1/2:
0.5084005792422687074591088492585899413195411256648216487244977963526253942287802426193842100493449551
5
-1/2:
-0.4926492299811078198961903994938682731492461804609724055239158196600472269993902882829523477323194209
6
1:
1.017343061984449139714517929790920527901817490032853561842408664004332182901957897882773977938535171
6
-1:
-0.9855510912974351040984392444849542614048856934693268880348333932541968021862717135739372911279559465
6
1/2:
0.5040953978039885506900465097888790952065222893266744492313452610836359208556675584443398687621171251
6
-1/2:
-0.4962516784926481541672722801416896854301368596152377931978220763356005761923953057452021357138301787
7
1:
1.008349277381922826839797549849796759599863560565238706417283136571601478317355735346096968913851324
7
-1:
-0.9925938199228302826704257131333936852311156924314068516295130875626702052186470519813142037745723970
7
1/2:
0.5020145633247084945674892956407070362801881524897887342721756595635450663591993196507832050490948478
7
-1/2:
-0.4981005683620026447703403364251222106001611381118260265970867421787712816145015139492729721488172954
8
1:
1.004077356197944339378685238508652465258960790649850020329110202652582952574748814395287230372371971
8
-1:
-0.9962330018526478992272892600828036178741251594728980670452890291943596482577585892828240488850878151
8
1/2:
0.5009966590970519105573559055302724587259558358886052175308664834176917877324304974002824107453998229
8
-1/2:
-0.4990416076565572282282818393382707895220364215406196012409718540438756660712833737042508251953860571
9
1:
1.002008392826082214417852769232412060485605851394888756548596615909785053390258398950393069127169586
9
-1:
-0.9980942975416053307677830318525979508743339535378774723433286603788874555254527020794930962008915800
9
1/2:
0.5004948881059536100404860164129523630702360512816971765254046602198027945619594500239891024068443212
9
-1/2:
-0.4995178456076264023814559297109300346211564802694674792346447755384725078147107232890640830899000720
10
1:
1.000994575127818085337145958900319017006019531564477517257788994636291465151912954397041968610385653
10
-1:
-0.9990395075982715656392218456993418314259296496668906471068948755061424583840381244079852460154434933
10
1/2:
0.5002463206060067750096752404960275553344113104371739078647247977641556184465001655592415520672407392
10
-1/2:
-0.4997579196224197507364661890406000936979154118119146444560890788195284312052553089324652301317417284
11
1:
1.000494188604119464558702282526469936468606435758208617119141436100054059798219814702591843023560630
11
-1:
-0.9995171434980607541440941748286901806712738122857884915164860245413626007554481156647963431768579339
11
1/2:
0.5001227915298679551870888668610375955949797622803764416033722177199941672308860250837730629387194903
11
-1/2:
-0.4998786210154795414898139297728608131322027919364474736466490144971062858545326705636522726374099973
12
1:
1.000246086553308048298637998047739670960416088458003404533040952133252019681940913049042808551900699
12
-1:
-0.9997576851438581908531796787127554230742049477898110200581713032308432052192055903375930806180374667
12
1/2:
0.5000612742269005787413337756585241938418918824697354470987583088892577588981301714400314286205440610
12
-1/2:
-0.4999391964493482330096933079719563864160341297491001493995301789591552268671168092171991614552446653
1/2
-1:
-0.6048986434216303702472659142359554997597625451302473803785466480821872534950603573274039569183495544
1/2
-1/2:
-0.3737522379809730564474733173828676216876933326443339375618518706722384733954146894626745259209314481
1/2
1/2:
0.8061267230428522613247578132154185232619146545083196780295787480193098234476541073498637068640752493
3/2
-1:
-0.7651470246254079453672687586034781795124679693458281781499491988342806728636621443796121478973476475
3/2
-1/2:
-0.4298873215805792677829217858907932221117458113994042892589872873817339845793204966285274610088703746
3/2
1/2:
0.6248370208199138536338193129462408798779654799583790826077318027103234267233902022699391490673624507
3/2
1:
2.612375348685488343348567567924071630570800652400063407573328248814927767688272860996243868126311952
5/2
-1:
-0.8671998890121841381913471776789571524992770777765975357717541216387952126134093086215179263627184846
5/2
-1/2:
-0.4622977821900634381891643232745464898869395902698162924524601235418487584476355832808822165731566009
5/2
1/2:
0.5549972787175122932105036190628745937248953841383270612692351047116564570758057526206734196974045757
5/2
1:
1.341487257250917179756769693348612136623037629505986511253796728340918923813185441581761085998697994
7/2
-1:
-0.9275535777739480351136094921918208473966372719921564313859481917054451443105954511930421014291858936
7/2
-1/2:
-0.4801763979926349820449798491598374740716918919053272926286545293889341149999949763037626937028950454
7/2
1/2:
0.5254123061416473462456725910643052589736550254149983566850931976101684316469996301779138002227702091
7/2
1:
1.126733867317056646427812491854984272221996957403602963842396038636678337584321046872404164158569965
Definition
The polylogarithm $\operatorname{Li}_s(x)$ is defined by $\sum_{n=1}^{\infty}x^n/n^s$ for $|x|<1$ and by its principal analytic continuation elsewhere [8]. This table gives real values at positive rational orders $s$ and rational arguments $x\leq1$, with $s>1$ at $x=1$.
Parameters
$s$
—   order of the polylogarithm ($s>0$)
$x$
—   argument ($x\leq1$, with $s>1$ if $x=1$)
Formulas
(1)
$\mathrm{Li}_s(x)=\sum_{n=1}^{\infty}x^n/n^s$ for $|x|<1$, and the series also converges at $x=\pm1$ when $s>1$.
(2)
$\mathrm{Li}_s(1)=\zeta(s)$ and $\mathrm{Li}_s(-1)=-(1-2^{1-s})\zeta(s)$.
(3)
For $0<x<1$, $\mathrm{Li}_2(x)+\mathrm{Li}_2(1-x)=\pi^2/6-\log(x)\log(1-x)$.
(4)
For $x>0$, $\mathrm{Li}_2(-x)+\mathrm{Li}_2(-1/x)=-\pi^2/6-\log^2(x)/2$.
(5)
$\mathrm{Li}_2(1/2)=\pi^2/12-\log^2(2)/2$, and $\mathrm{Li}_3(1/2)=7\zeta(3)/8-\pi^2\log(2)/12+\log^3(2)/6$.
(6)
$\mathrm{Li}_2(1/3)-\mathrm{Li}_2(1/9)/6=\pi^2/18-\log^2(3)/6$.
Comments
(7)
The case $s=2$ is the dilogarithm, also called Spence's function, and $s=3$ is the trilogarithm. The name Jonquière's function is also used for the polylogarithm [1].
(8)
For the listed arguments, the defining series gives the real value and no analytic continuation is needed. Arguments $x>1$ are complex on the principal branch. For $x<-1$, the principal-branch values are real; when $s=2$, applying (4) with $-x>1$ expresses $\mathrm{Li}_2(x)$ in terms of $\mathrm{Li}_2(1/x)$ and $\log(-x)$.
(9)
The endpoint $x=1$ gives the Riemann zeta values by (2). The endpoint $x=-1$ gives $-\eta(s)$, where $\eta$ is the Dirichlet eta function. The argument $x=0$ is omitted because $\mathrm{Li}_s(0)=0$.
(10)
Integer orders give a weight-indexed polylogarithm family, with zeta and alternating-zeta endpoints [8]. The half-argument values, their negative partners, and the low-weight reflection and Landen partners broaden recognition beyond the few reducible constants. Half-integer orders occur in Fermi-Dirac and Bose-Einstein integrals [8]; their selected fugacity arguments here are bounded reference choices, not a claim of a specific experiment or quantile. The negative arguments use the real principal continuation; the singular endpoint $x=1$ is included only for $s>1$. The finite order cutoffs and the extra simple low-weight arguments are explicitly exploratory, and are not a denominator-complete rational mesh.
Programs
(P1)
Python
from balanced_values import BalancedValues
from numberdb._write import to_text
reference = BalancedValues()
for parameters in reference.enumerate():
    print(parameters, to_text(reference.value(parameters, reference.digits), reference.digits))
Links
Similar tables
Values of the Riemann zeta function at rational numbers —   holds the $x=1$ endpoint values in (2)
Multiple zeta values of length 2 —   multiple zeta values are multiple polylogarithms at argument $1$
Multiple zeta values of length 3 —   multiple zeta values are multiple polylogarithms at argument $1$
Hyperbolic volumes of the prime knots with at most ten crossings —   the hyperbolic volumes are sums of Bloch-Wigner dilogarithms, which are built from $\mathrm{Li}_2$
Values of Dedekind zeta functions of cubic fields at 2 —   its formulas use Zagier's theorem expressing the values through dilogarithms
Values of the Clausen functions at rational multiples of $\pi$ —   gives real and imaginary parts of the same polylogarithm on roots of unity
Data properties
Entries are of type: real number
How well the digits are known: proven
How they were obtained:

All 85 stored entries are recomputed from exact rational or enclosed square-root inputs using Arb at 100 decimal digits plus 160 guard bits. Existing numeric strings and row annotations are retained unchanged; their written intervals strictly contain the fresh enclosures. Exact anchors and rational polynomial integrals use exact arithmetic.

more

Every inexact result is finite, with at least 409 relative accuracy bits. Independent mpmath computations at 150 and 200 decimal digits check every value; the AGM check uses its elliptic-integral expression, and the incomplete-gamma check uses integer/half-integer closed forms. Known nonzero and exact controls, shifted-last-place rejection controls, enumeration checks and applicable identities pass. The attached balanced-evidence.json records the checks, and balanced_checks.py repeats them. The current generate.py and balanced-selection.json reproduce the full retained-plus-added selection; older auxiliary attachments are preserved as historical sources, not claimed as the default enumeration.

Table is complete: no (it preserves the previous 13 values and includes integer weights $2\leq s\leq12$ at $x=1,-1,1/2,-1/2$; weights $2,3,4$ also have $x=1/3,2/3,1/4,3/4,-1/3,-1/4,-2,-3$; orders $s=1/2,3/2,5/2,7/2$ have $x=-1,-1/2,1/2$ and, for $s>1$, $x=1$; the zero-argument anchor at weight 2 is exact, and the earlier dilogarithm at $1/9$ is retained)


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