Values of the Lambert $W$ function

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Numbers

branch

$x$

value

$W_0(x)$

1/6:

0.1442749507208862235033084788162594402003809667604615214817757597058134289661129894356917033092974868

$W_0(x)$

1/5:

0.1689159734991095651164749037058183987284469135107297553319388783258832067427865597883933228919370041

$W_0(x)$

1/4:

0.2038883547022401644431818313271398701493524772101596349734062600818193640670940526181645713107956088

$W_0(x)$

1/3:

0.2576276530497367042829162016260977909096926475032044915339511440663191292752043724596398879341002506

$W_0(x)$

2/5:

0.2971677506731385467797269622470213419044581015501274519083671027242648030351552074110959032585686367

$W_0(x)$

1/2:

0.3517337112491958260249093009299510651714642155171118040466438461099606107203387108968323038321915693

$W_0(x)$

3/5:

0.4015636367870725946626664537946836866230027160063511402379921560604975325206585381386527082762303647

$W_0(x)$

2/3:

0.4325627555319995690814797979265677452101240875642807778567657344550202638958385992478817559918364435

$W_0(x)$

3/4:

0.4691502106949882333210584590439702658499137457474476102069630093497940448447593361363013390517535299

$W_0(x)$

4/5:

0.4900678588015798580540931036905853669051863381329102981298158915386428548921964923481760319517146305

$W_0(x)$

5/6:

0.5036172222768952727976523382038810330533076531390096538396119046209524446620916135900716322850891587

$W_0(x)$

0.5671432904097838729999686622103555497538157871865125081351310792230457930866845666932194469617522946

comment: This is the omega constant.

$W_0(x)$

7/6:

0.6246747863453050609465602030202328250548126768278239377528355187673198729514798352829479219054195475

$W_0(x)$

6/5:

0.6355640163648697958889748066623403928117456956262446170560543221703166285772565843404568397853148396

$W_0(x)$

5/4:

0.6515479054418520968410021182981441989561981947219879836337783906117838993046201495139126021744840092

$W_0(x)$

4/3:

0.6773093057811563054970306649286917514150839315767090074035756532666502335475773304947978373881524795

$W_0(x)$

7/5:

0.6971815973648547772846301509091665039804028644773350951777460938619895189399516423788383362333265924

$W_0(x)$

3/2:

0.7258613577662262570486893992763068797061628495042869835157402542862628476329767313550440204916875257

$W_0(x)$

8/5:

0.7532980036023616390553304809419867528048921677553178959600549686745116984233991384726227581338034326

$W_0(x)$

5/3:

0.7709532136765081509324749582617279732104196074031655279444406919430964963991457586004382603471644205

$W_0(x)$

7/4:

0.7923578925895097685675184150114791737680386117487873599640231703742031261984480139139922295177776230

$W_0(x)$

9/5:

0.8048660624091565353931191694296048706768475279101851354251045427765461714112210437537376668047650076

$W_0(x)$

11/6:

0.8130717381233479346752271276137620174088034601988036563214828810115938968918678270452773864425307384

$W_0(x)$

0.8526055020137254913464724146953174668984533001514035087721073946525150656742630448965773783502494847

$W_0(x)$

13/6:

0.8898698976221504860967310768520557195305220691552961558589956814251055796848053904896677305617710169

$W_0(x)$

11/5:

0.8970741340486472927982531353046667717588035992173777886501529559971987637972632887179601954203399499

$W_0(x)$

9/4:

0.9077340528432470162770423323986889103280977102137643275475025454830898914708257041813734637782417292

$W_0(x)$

7/3:

0.9251246520082482188273946485746896777959602817107647726485633976515491015639418435635719610891579400

$W_0(x)$

12/5:

0.9387135946623280212308250778768483202247649374486198969761335626714861544269088900483077044007045121

$W_0(x)$

5/2:

0.9585863567287029121698667813324521409753469088398610136783799643687560278216210982375841408159759324

$W_0(x)$

13/5:

0.9778798817817953104139510425476144143691104137446648034461410477499447669208664052974838299194530518

$W_0(x)$

8/3:

0.9904376330033606049562782872308194318726520526581744637237028085364009884180486875411832021034641118

$W_0(x)$

11/4:

1.005808859023790603619943968405326772958133990387450350900042972644043423351163898873117936109225687

$W_0(x)$

14/5:

1.014864404871308908211893902779854112565930977807983431231045965891727976715882572558728710911721019

$W_0(x)$

17/6:

1.020833966944899824782492299494794604662633230285467546303482982671358518990850356729059131089952750

$W_0(x)$

1.049908894964039959988697070552897904589466943706341452932871583316649050444442957885678666822434674

$W_0(x)$

19/6:

1.077777997707128271678772715501599262490772371141406379230076965578114616688407706368955173026244846

$W_0(x)$

16/5:

1.083216216147652079942637864129349989111654365509911057873243242833020424799519826526740712227600409

$W_0(x)$

13/4:

1.091292355331711217610315153332573107962749822840720679057316659195037119617688046321894486987700807

$W_0(x)$

10/3:

1.104542018324511726887586471140460074231215374219101174205425381499195249875079372369874814664948368

$W_0(x)$

17/5:

1.114958366694555942827881488713403275674395195771797552355278535347598006594278939081892540512803244

$W_0(x)$

7/2:

1.130289326974135826518558809126463513693522502608508510883182944218668230475865302768366189956980177

$W_0(x)$

18/5:

1.145282507472244320134612952998330869128913079421793989160191809626496425447288207433703418724602285

$W_0(x)$

11/3:

1.155097890268583833715229863847872239067469274209623251609000131745874581370467310739014565330907615

$W_0(x)$

15/4:

1.167172059822758442561268071874453654625699916966151656244031434448975225248881737112367349327855065

$W_0(x)$

19/5:

1.174315576734081303501386207133154680896524283970323628938154228299891289802526696302653569515415866

$W_0(x)$

23/6:

1.179036861001918647755500948466312524307984373289575236896571407815208174475593641852874804849671257

$W_0(x)$

1.202167873197042939212074165495153447501512521829625981739203590700634132981772677227826104976568377

$W_0(x)$

25/6:

1.224546106010654210076548735855357603126281891565571930694416675066710236727800236541587150005271730

$W_0(x)$

21/5:

1.228935873694932522778030316755271300788086434787682265336014507931170265257205324477792059932159813

$W_0(x)$

17/4:

1.235468628531174065493546395946862694707824503570995185520695779888249640243914121721554594458201436

$W_0(x)$

13/3:

1.246221164544739520862691445937666985250447697013701526823294500227169866211756559593325447564681333

$W_0(x)$

22/5:

1.254704474531127399394401695738510748420861053410644347296477924707399073850081356138705713900945725

$W_0(x)$

9/2:

1.267237814307434761430955213676618689673645294647245903242830561034682709534268052745282826690204790

$W_0(x)$

23/5:

1.279548793099314277898783546883738139725380143694633088174406114371279868884444224719422698058154928

$W_0(x)$

14/3:

1.287636597710206078859431200980898071238609825767889254964549826693509943612354711671344528348664114

$W_0(x)$

19/4:

1.297615934084622914396998480623210853588158620235297443094510021093784729905157933053653467622741835

$W_0(x)$

24/5:

1.303535629884274840134037179665111345274656370731951203647830112081851277484604328487444818120430284

$W_0(x)$

29/6:

1.307454354742013742832699504341987259741516800671598709971588434988125179381027670498867666686635845

$W_0(x)$

1.326724665242200223635099297758079660128793554638047479789290393025342679920536226774469916608426789

$W_0(x)$

31/6:

1.345478226593652143616923935371744407557966356072601056749667784089246547834108387437649047769500326

$W_0(x)$

26/5:

1.349169446948141090367605016433759162796654676059365303860215139534893916543335510724466307265690082

$W_0(x)$

21/4:

1.354670111822578873479154223881942243526274759503976501884011943362603151392883531256315063543093274

$W_0(x)$

16/3:

1.363743177885866250459164034393021210389464551511956733674767597127218974926390393101620148386481375

$W_0(x)$

27/5:

1.370918211129932770755366721432791989165678227349311940132254640314527541589434597798131860886895586

$W_0(x)$

11/2:

1.381545379445041311675208005914311971044238578990888587233615982811868047391200294619627558246147215

$W_0(x)$

28/5:

1.392014569377106858877757327014387558800263740885058281398928786183195966746934975031027968352018575

$W_0(x)$

17/3:

1.398908654943390370055627475596431272589148584371693317033297669725548380372443082969609277296646932

$W_0(x)$

23/4:

1.407432635806441921698265721193731556441435017818554897914483339270156503499291079144658023713175828

$W_0(x)$

29/5:

1.412498087498320935750810640371702734833040823753211479037661107858913848922961228551534318252165701

$W_0(x)$

35/6:

1.415855001971021168951293812930347581069675174381697063872010071910688487367815709140461951387172907

$W_0(x)$

1.432404775898300311234078007212058694786434608804302025655948496343399593259831116857638422299445651

$W_0(x)$

37/6:

1.448576851012758523216806518098818720220669837657706663705039146140906634675228418585348062900154072

$W_0(x)$

31/5:

1.451767507364093078157608062765648371415645558894461719837739201490081964706304288021998270580791932

$W_0(x)$

25/4:

1.456526779974448398096883412307231533886200382680868861698255507747571433922953849281171897508991357

$W_0(x)$

19/3:

1.464388762235657855125371883425357121558588971415724273019277020665917448701187421329930342611518872

$W_0(x)$

32/5:

1.470616374843280500888942581102091678235170339542990079312376184303622815455986216850166156645188612

$W_0(x)$

13/2:

1.479856830173851384350080235844579087363188967148783629001011031793289914980806643895523982901465796

$W_0(x)$

33/5:

1.488978999315075661283091801156075807492844548163511305339387567153047276749055058491978767778277778

$W_0(x)$

20/3:

1.494996271815342603038114862572159981799565842881615191085894515371826867260225264649170122164509044

$W_0(x)$

27/4:

1.502447235726658434985624166365311982597727786955237156901161045431199582243738312130240737833117103

$W_0(x)$

34/5:

1.506880795919067780384000668931621512364519468662284145263021595047341727228883451972392856440175215

$W_0(x)$

41/6:

1.509821298811023327715070126224421528189598072192511130046530467924930350304657704434050218529121336

$W_0(x)$

1.524345204984144369122476060675779385408719418799609942927160045982360820247712017874947206364556560

$W_0(x)$

43/6:

1.538580444460415910560234421671430182356090617410978883494572488155025763265503276629820666384987503

$W_0(x)$

36/5:

1.541393893394668909773145288225234829078998364763268314758523772189108060886503191425595770167928533

$W_0(x)$

29/4:

1.545593493859050661419384273899133267871439542231630003374987650591291207592347254887081465064644872

$W_0(x)$

22/3:

1.552538701605226696645671899898275666665570207575080155895785774286389003409571843075820355913989616

$W_0(x)$

37/5:

1.558046930769876005573372541516796052256372941575890720113229260166716850885939091181975480652863187

$W_0(x)$

15/2:

1.566230953782387539419855977537184067357969192716524066571934108030291141597762205879293233966151821

$W_0(x)$

38/5:

1.574322944161214202605425106651420141991308339649643036496408714872871100522240466523314505302209351

$W_0(x)$

23/3:

1.579667527804741063232867263945130661283891576311703241042949457884692625213971108338123667772782250

$W_0(x)$

31/4:

1.586292997379286044777418860284980235870406134896191294426658405716320928553722440956240536216820847

$W_0(x)$

39/5:

1.590239254349745639811200233946952215989852683052336154625365502828687993715187414194917327556605252

$W_0(x)$

47/6:

1.592858150825894833143991167601775288379371025860185078169683421714577194352379714631874451626548032

$W_0(x)$

1.605811996320177596033746190198001992299603560529966120743330224549154103132734177725394347125840549

$W_0(x)$

49/6:

1.618537725711416668946159746252877169673225125712433900886684152597179599108616529504115914032074176

$W_0(x)$

41/5:

1.621056226075401344113319257715062378136306176975137329909669257466505812903942756881597364299532326

$W_0(x)$

33/4:

1.624817622915423750697900968950273530375834521784459195873060072031579229205131396451920735841149701

$W_0(x)$

25/3:

1.631043526137634479000220174586580080696864302671387723927276606447165692966547852693637106762472740

$W_0(x)$

42/5:

1.635986015616761312690976856493906815897644085604416666809959794035801124021074336101392138171789906

$W_0(x)$

17/2:

1.643337144776101374502357423646068208041713986273243320970330946711671992126946523878600082632458009

$W_0(x)$

43/5:

1.650614537636023766787119056671204324361398436651539015713366198446109954931717203431561493715318292

$W_0(x)$

26/3:

1.655425920099590646555981110453120518484121005096298772630286102555958136334201783164828695313802278

$W_0(x)$

35/4:

1.661395680104097551752311165262197268441381301212118505099142870184452067001391035588189516079615048

$W_0(x)$

44/5:

1.664954141233666371146025987887147610877377146063155119506993599434376040760644968511958568616430837

$W_0(x)$

53/6:

1.667316810388706558428435971698422496525956612025652552710540918322599007838318532432834651653067576

$W_0(x)$

1.679016419785598195445932780030038598991102566022662384061271245995063940097862328659342253751310883

$W_0(x)$

55/6:

1.690531022140238680422635143895657990077268041794564301480593985297128592706667447034232727390962182

$W_0(x)$

46/5:

1.692812271919545843630764275117986026149602694688884601146617598192057060739035739098164604097289580

$W_0(x)$

37/4:

1.696220821357790402865666441847657564005325964636608369346311515763757468865870883765936591386717778

$W_0(x)$

28/3:

1.701866582870916252087074411362165107409450728192875302044222688460940551313081879010153480207118869

$W_0(x)$

47/5:

1.706351956427362209881664556641568362730503078500278299850826287646390772419899766722138024018137889

$W_0(x)$

19/2:

1.713028779034963538666375431375159970564598382475255663236495655294380436482120173619256403302202938

$W_0(x)$

48/5:

1.719645141825824035195682210056562333354626194320637200908288813450861760567287130608436946068355528

$W_0(x)$

29/3:

1.724023017783469173257917537237721728437570669952764304484464896821391913223125289154373363102578398

$W_0(x)$

39/4:

1.729458773268711137403121612250985107587421406162449103734906876803979676710192800427721624931541462

$W_0(x)$

49/5:

1.732700951182371204928614817643603248348124315213301281530864759622580688536611021785355466200671675

$W_0(x)$

59/6:

1.734854453354289290767547771097659837188149677009512153397329187256541883710123462732383830149290868

$W_0(x)$

10:

1.745528002740699383074301264875389911535288129080941331322206048555557259941551704989523510778883075

$W_0(x)$

-1/10:

-0.1118325591589629648335694568202658422726453622912658633296897727621943319600088273854870109175450158

$W_0(x)$

-1/9:

-0.1260358732691560480733005507016763434924663343345936748965181034019731111368490928445619134165148355

$W_0(x)$

-1/8:

-0.1444213531375097291689673973929506964945240188378136452921781603535255447194649839438887511551181045

$W_0(x)$

-1/7:

-0.1691926038562251029660671049831814435964394192529518648529863026037438278908130553175907447244744749

$W_0(x)$

-1/6:

-0.2044814493399155336177577545103568904881569994079249050071973880762179473713908151298328671141408253

$W_0(x)$

-1/5:

-0.2591711018190737450566519502154067057135883397008937032218391275127128217957580326598838960236799743

$W_0(x)$

-2/9:

-0.2999551794476694433520294327069701613308368903933171328891190564233724856079021343448391693934499764

$W_0(x)$

-1/4:

-0.3574029561813889030688111040559047533165905550760120436276204485896714025961457962896168513444411851

$W_0(x)$

-2/7:

-0.4465426855126359478555268284833324547177862099710066716309234596653439811853604343150545499437416551

$W_0(x)$

-3/10:

-0.4894022271802149690362312519962933689234100060163590345114659679736814083816206187318524147462752111

$W_0(x)$

-1/3:

-0.6190612867359451121523269940209222333014717772629693524598360744929373522550887346110469261882588407

$W_{-1}(x)$

-1/10:

-3.577152063957297218409391963511994880401796257793075923683527755791687236350575462861463655620846808

$W_{-1}(x)$

-1/9:

-3.429696289158993827431388653314195123876826224541773596010263214749247518175935754955652063414017934

$W_{-1}(x)$

-1/8:

-3.261685684576488776905662364308739731721145393347809522040218079880635146768556728409460458942611242

$W_{-1}(x)$

-1/7:

-3.066421345069269412410724506976364909123882310233083793276489033138648112911014247276124956339615294

$W_{-1}(x)$

-1/6:

-2.833147892049342142611674642343132564014684277147565123233727179328355819234045938602374889059888660

$W_{-1}(x)$

-1/5:

-2.542641357773526424293806156661848290161474907529431767116934699793352535483158781587362627961593449

$W_{-1}(x)$

-2/9:

-2.364749488648077562062221320093018152773078682575886686834866711672991385616174968212803058936366609

$W_{-1}(x)$

-1/4:

-2.153292364110349649169099150092981375536206485319477695884511507721362584650649378945727909538027439

$W_{-1}(x)$

-2/7:

-1.888597412029828149918298335177048198356094333854986072511936800169277139419382516462227033551784252

$W_{-1}(x)$

-3/10:

-1.781337023421627611974170281512745260821558356454461408571419292426296682540163764423775917550185065

$W_{-1}(x)$

-1/3:

-1.512134551657842473896739678072038704603650385135359454259285473998977160511574827324265488152779835

Definition

The Lambert $W$ function is the multivalued inverse of $w\mapsto we^w$ [2] [1]. This table gives real values of the principal branch $W_0(x)$ and the lower real branch $W_{-1}(x)$ at rational arguments where those branches are real.

Parameters

branch

— branch of $W$ ($0$ is the principal branch $W_0$ and $-1$ is the lower real branch $W_{-1}$)

$x$

— argument ($x\in\mathbb{Q}$; for branch $0$, $x>-e^{-1}$; for branch $-1$, $-e^{-1}<x<0$)

Formulas

(1)

$W_k(x)e^{W_k(x)}=x$ [1].

(2)

If $y\geq-1$, then $W_0(ye^y)=y$. If $y\leq-1$, then $W_{-1}(ye^y)=y$ [1].

(3)

The omega constant is $\Omega=W_0(1)$ and satisfies $e^{-\Omega}=\Omega$ [3] [5].

Comments

(4)

The branch convention is that of [1]. On $x\geq0$ the principal branch is the nonnegative real solution. On $-e^{-1}<x<0$, the principal branch satisfies $-1<W_0(x)<0$, while the lower real branch satisfies $W_{-1}(x)<-1$. The other branches are complex and are not stored here.

(5)

Wolfram Language calls the same function ProductLog [4].

(6)

At the branch point $x=-e^{-1}$, both real branches have value $-1$, and $W_0(e)=1$; neither argument is rational.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.rational_field import QQ

CBF = ComplexBallField(numberdb.bits(100, losing=64))

CBF(QQ(1)).lambert_w(0).real()
CBF(-QQ(1) / QQ(4)).lambert_w(-1).real()

(P2)

Python

import mpmath

mpmath.mp.dps = 100
mpmath.lambertw(1, 0)
mpmath.lambertw(mpmath.mpf(-1) / 4, -1)

Links

[1]

DLMF 4.13: Lambert W-Function

[2]

Wikipedia: Lambert W function

[3]

Wikipedia: Omega constant

[4]

MathWorld: Lambert W-Function

[5]

OEIS A030178: decimal expansion of the omega constant

Similar tables

Euler's constant $e$ — the real branch point is at $x=-e^{-1}$, where both real branches have value $-1$, and the principal branch also satisfies $W_0(e)=1$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds $W_0(x)$ for every positive rational $x=a/b$ in lowest terms with $b\leq6$ and $x\leq10$, and it holds both $W_0(x)$ and $W_{-1}(x)$ for every negative rational $x=a/b$ in lowest terms with $b\leq10$ and $-e^{-1}<x<0$; the exact value $W_0(0)=0$ is omitted)

How they were obtained:

Each value was computed as a Sage complex ball with arb at numberdb.bits(digits, losing=64) bits, then returned as a real ball after checking that its imaginary part contained zero.

The entries were checked against mpmath at 150 decimal digits, against the defining equation in (1), against the branch inequalities in the branch comment, and against the OEIS value for $\Omega=W_0(1)$ [5]. The widest returned ball had radius less than $10^{-118}$ for 100 requested digits.