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Values of the Beta function $B(a,b)$ at pairs of rational numbers
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Numbers
$a$
$b$ 
1
1:
1
1
2:
1/2
1
3/2:
2/3
1
4/3:
3/4
1
5/3:
3/5
1
5/4:
4/5
1
7/4:
4/7
1
6/5:
5/6
1
7/5:
5/7
1
8/5:
5/8
1
9/5:
5/9
1
7/6:
6/7
1
11/6:
6/11
2
2:
1/6
1/2
1:
2
1/2
2:
4/3
1/2
1/2:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
comment: $\pi$.
1/2
3/2:
1.570796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534
comment: $\pi/2$.
1/2
2/3:
2.587109559229790534953515025131211637657858514840401576355253541526067494556044450440644287884620027
1/2
4/3:
1.682618526390545113410022894860352962555626646508783315418392784491541719178163955285022719782029742
1/2
5/3:
1.478348319559880305687722871503549507233062008480229472203002023729181425460596828823225307362640015
1/2
3/4:
2.396280469471184414879844984560647756454425326431303116527349905892810428287831341671771112979586779
1/2
5/4:
1.748038369528079873643226393260746275788503300954415441877881135867193724713609500153530355307619396
1/2
7/4:
1.437768281682710648927906990736388653872655195858781869916409943535686256972698805003062667787752067
1/2
3/5:
2.774501918484055737859139776264637993504487999315072019106777588813259132137934804168429095790271355
1/2
4/5:
2.299287818447969763775464390810928981569915266105158908237410638236132556269677085767149539015020761
1/2
6/5:
1.791043749738867524164975305761140105839463143245018757568972803574785493143363131735902987486939483
1/2
7/5:
1.635152880180391476323215166712734847888820641229988416710427462118238230768911470762865618439831333
1/2
8/5:
1.513364682809484947923167150689802541911538908717312010421878684807232253893418984091870415885602557
1/2
9/5:
1.414946349814135239246439625114417834812255548372405481992252700453004650012108975856707408624628161
1/2
5/6:
2.240502600666560439310412641668570469175075971041132013745100731380388730087870663240496817718391174
1/2
7/6:
1.821487985915686208864956267335698436416276133656181955812472064104054265521326880841004545316626583
1/2
11/6:
1.400314125416600274569007901042856543234422481900707508590687957112742956304919164525310511073994484
3/2
2:
4/15
3/2
3/2:
0.3926990816987241548078304229099378605246461749218882276218680740384770507857761248285043531677646335
3/2
5/3:
0.3411573045138185320817822011162037324383989250338991089699235439375034058755223451130519940067630805
3/2
7/4:
0.3195040625961579219839793312747530341939233768575070822036466541190413904383775122229028150639449038
3/2
8/5:
0.3603249244784487971245636073070958433122711687422171453385425440017219652127188057361596228299053707
3/2
9/5:
0.3075970325682902694013999185031343119157077279070446699983158044463053586982845599688494366575278610
3/2
11/6:
0.3000673125892714874076445502234692592645191032644373232694331336670163477796255352554236809444273894
1/3
1:
3
1/3
2:
9/4
1/3
1/2:
4.206546315976362783525057237150882406389066616271958288545981961228854297945409888212556799455074354
1/3
3/2:
2.523927789585817670115034342290529443833439969763174973127589176737312578767245932927534079673044612
1/3
1/3:
5.299916250856349871941068498945316107715605614607672590380715725506359005184323740816460980000150762
comment: The real period of the Dixonian elliptic functions.
1/3
2/3:
3.627598728468435701188156515284311464568132496185481151139769870776246362252707767368249976424120338
comment: $2\pi/\sqrt{3}$.
1/3
4/3:
2.649958125428174935970534249472658053857802807303836295190357862753179502592161870408230490000075381
1/3
5/3:
2.418399152312290467458771010189540976378754997456987434093179913850830908168471844912166650949413558
1/3
3/4:
3.425717493623563760877733540324068916774903908156820433415447844601497731665080183601471538941695987
1/3
5/4:
2.722965636665752084512338130864465417759137202089161206603166409543915423356461641251008901848846303
1/3
7/4:
2.371650572508621065223046297147432326998010397954721838518386969339498429614286280954864911575020299
1/3
2/5:
4.760547281926408006308053407067055555432512907306702929499095404890450570790339966861985980650831243
1/3
3/5:
3.824369802836423166888948072004630788960271099895487665662200336058080567725901856227745459068336646
1/3
4/5:
3.322418287153612332589998986164148511636730717055268875792265956006834893536189090059691437789394103
1/3
6/5:
2.770694947350024886356412645213976202004852778204260836361173856985573446091715926384151847957661032
1/3
7/5:
2.596662153778040730713483676582030302963188858530928870635870220849336674976549072833810534900453405
1/3
8/5:
2.458523444680557750142895189145834078617317135647099213639985930323051793538079764717836366543930701
1/3
9/5:
2.345236437990785175945881637292340125861221682627248618206305380710706983672604063571546897263101720
1/3
5/6:
3.259553792057860241716369771803679386833179762550141933734395414602608122091847023877112762613558265
1/3
7/6:
2.804364210650908522350038158100588270926044410847972192363987974152569531963606592141704532970049569
1/3
11/6:
2.328252708612757315511692694145485276309414116107244238381711010430434372922747874197937687581113047
2/3
1:
3/2
2/3
2:
9/10
2/3
3/2:
1.108761239669910229265792153627662130424796506360172104152251517796886069095447621617418980521980012
2/3
2/3:
2.053390217939177181033230936599038986867051426207190851764616838118017065439715039330353109430046603
2/3
4/3:
1.209199576156145233729385505094770488189377498728493717046589956925415454084235922456083325474706779
comment: $2\sqrt{3}\,\pi/9$.
2/3
5/3:
1.026695108969588590516615468299519493433525713103595425882308419059008532719857519665176554715023302
2/3
3/4:
1.871846848991417524656833363903755848005048159218010638556115605776605728007536469988892871315072971
2/3
5/4:
1.268494941193114679216990481305249431341768434541988664757207638156036373483799730358157059815268043
2/3
7/4:
0.9909777435836916307006764867725766254144372607624762204120612030582030324745781311705903436373915728
2/3
4/5:
1.780125734852577484400207974363248015597093597124749432023582293369273075749536748193751987208778487
2/3
6/5:
1.307671663451412030210039681855878708427163907186046642684116873021841143727554107763147232768562050
2/3
7/5:
1.166418696059919770595568533232203585831187801158286786048692749512025708863185388645145880960458114
2/3
8/5:
1.057668735482964062161160343205887587625514285254436024176295794351634518080284907066260171754901326
2/3
9/5:
0.9709776735559513551273861678344989175984146893407724174674085236559671322270200444693192657502428110
2/3
5/6:
1.724739706153193689969010016754141091771905676560267717570169027684044996370696300293762858589746685
2/3
7/6:
1.335495209426766461474399634822037652990816251192373062091905958975772395650825992382590695335110040
2/3
11/6:
0.9581887256406631610938944537523006065399475980890376208723161264911361090948312779409793658831926025
4/3
2:
9/28
4/3
3/2:
0.4588959617428759400209153349619144443333527217751227223868343957704204688667719878050061963041899295
4/3
4/3:
0.5299916250856349871941068498945316107715605614607672590380715725506359005184323740816460980000150762
4/3
5/3:
0.4030665253853817445764618350315901627297924995761645723488633189751384846947453074853611084915689264
4/3
7/4:
0.3794640916013793704356874075435891723196816636727554941629419150943197487382858049527783858520032478
4/3
7/5:
0.4993581064957770635987468608811596736467670881790247828145904270864108990339517447757327951731641164
4/3
8/5:
0.4238833525311306465763612395079024273478132992494998644206872293660434126789792697789373045765397760
4/3
9/5:
0.3664431934360601837415440058269281446658158879105075965947352157360479661988443849330542026973596437
4/3
11/6:
0.3581927244019626639248757990993054271245252486318837289818016939123745189111919806458365673201712379
5/3
2:
9/40
5/3
5/3:
0.2933414597055967401476044195141484266952930608867415502520881197311452950628164341900504442042923719
5/3
7/4:
0.2733731706437770015726004101441590690798447615896486125274651594643318710274698292884387154861769856
5/3
9/5:
0.2624263982583652311155097750904051128644364025245330858020023036908019276289243363430592610135791381
5/3
11/6:
0.2555169935041768429583718543339468284106526928237433655659509670643029624252883407842611642355180273
1/4
1:
4
1/4
2:
16/5
1/4
1/2:
5.244115108584239620929679179782238827365509902863246325633643407601581174140828500460591065922858187
comment: $2\varpi=\Gamma(1/4)^2/\sqrt{2\pi}$, where $\varpi$ is the lemniscate constant. OEIS [4] uses this value as its lemniscate constant.
1/4
3/2:
3.496076739056159747286452786521492551577006601908830883755762271734387449427219000307060710615238791
1/4
1/3:
6.353586485553421530528788972017085974771320138208042815407388288935802654498410496252354104313974706
1/4
2/3:
4.651148117708087157128965098119247914919817593320625104109761339905466702773932344646575885989316157
1/4
4/3:
3.630620848887669446016450841152620557012182936118881608804221879391887231141948855001345202465128404
1/4
5/3:
3.382653176514972477911974616813998483578049158778636439352553701749430329290132614288418826174048114
1/4
1/4:
7.416298709205487673735401388781040184870395294087067622343712180224087107352479913429087446601487589
comment: $\Gamma(1/4)^2/\sqrt{\pi}$.
1/4
3/4:
4.442882938158366247015880990060693698614621689375690223085395606956434793099473910575326934764765237
1/4
5/4:
3.708149354602743836867700694390520092435197647043533811171856090112043553676239956714543723300743795
1/4
7/4:
3.332162203618774685261910742545520273960966267031767667314046705217326094824605432931495201073573928
1/4
2/5:
5.807488203919792232040195482993956046462769254161563319704429598663137515537024740466896293361908708
1/4
3/5:
4.853311552538344457535257142392975951407856526304396824346769136285895066070442676229995652991602801
1/4
4/5:
4.335926338328535230714245351302282240576653171330930443382812078839983852513965551471461555156406516
1/4
6/5:
3.758684485988704430467805766027459901960407408681038855292317735008014875588375193316299785069947683
1/4
7/5:
3.573838894719872142793966451073203720900165694868654350587341291485007701868938301825782334376559205
1/4
8/5:
3.425866978262360793554299159336218318640839900920750699538895860907690634873253653809408696229366683
1/4
9/5:
3.303562924440788747210853600992215040439354797204518433053571107687606744772545182073494518214404964
1/4
5/6:
4.270690375575460918332119732263871837161613245759226622372201266094244509483879423909023231778711383
1/4
7/6:
3.794265421319873843023941560271547678952295958828461100116564259954251112821616626410008514764554410
1/4
11/6:
3.285146442750354552563169024818362951662779419814789709517077896995572699602984172237710178291316449
3/4
1:
4/3
3/4
2:
16/21
3/4
3/2:
0.9585121877884737659519379938242591025817701305725212466109399623571241713151325366687084451918347115
3/4
4/3:
1.054066921114942695654687243176636589776893510202098594897060875261999302050793902646606627366675688
3/4
5/3:
0.8808691054077258939561568771311792225906108984555344181440544027184026955329583388183025276776813981
3/4
3/4:
1.694426169587958173212998246964383272962891820653884370121158745319468009668269519446400587989222460
3/4
5/4:
1.110720734539591561753970247515173424653655422343922555771348901739108698274868477643831733691191309
3/4
7/4:
0.8472130847939790866064991234821916364814459103269421850605793726597340048341347597232002939946112299
3/4
4/5:
1.605036944144491960095517286889063343837778864099608959108379548876322162781261602822798799023535736
3/4
6/5:
1.148241177287136301579131296657190006922874361401864868444122967724018798307380269271948906159192055
3/4
7/5:
1.013299971711377450611317185112599143114188575581151309358838884711865636263844930289710629452478990
3/4
8/5:
0.9101239232353067244973430049099084232086704646054331781375813758973695056374543675646202055193286946
3/4
9/5:
0.8284061647197377858557508577491939839162729621159272047010991220006824065967801821020897027218248961
3/4
5/6:
1.551151201973352019621419522311481731448820521397438955132592129175619984162080404110463470866313684
3/4
7/6:
1.174928847663104236279532811220401765500451967097984751264297570588708435730518282713405408163398170
3/4
11/6:
0.8163953694596589576954839591113061744467476428407573448066274364082210442958317916370860372980598336
5/4
2:
16/45
5/4
3/2:
0.4994395341508799638980646837887846502252866574155472691079660388191982070610312857581515300878912559
5/4
4/3:
0.5732559235085793862131238170240979826861341478082444645480350335881927207066235034212650319681781690
5/4
5/3:
0.4412156317193442362493879934974780630753977163624308399155504828368822168639303409941415860227019279
5/4
5/4:
0.6180248924337906394779501157317533487391996078405889685286426816853405922793733261190906205501239658
5/4
7/4:
0.4165202754523468356577388428181900342451207833789709584142558381521657618530756791164369001341967410
5/4
7/5:
0.5414907416242230519384797653141217758939644992225233864526274684068193487680209548220882324812968492
5/4
8/5:
0.4629549970624811883181485350454349079244378244487500945322832244469852209288180613255957697607252274
5/4
9/5:
0.4028735273708278960013236098770993951755310728298193211040940375228788713137250222040846973432201176
5/4
11/6:
0.3942175731300425463075802829782035541995335303777747651420493476394687239523581006685252213949579738
7/4
2:
16/77
7/4
7/4:
0.2541639254381937259819497370446574909444337730980826555181738117979202014502404279169600881983833690
7/4
9/5:
0.2436488719763934664281620169850570540930214594458609425591468005884360019402294653241440302123014400
7/4
11/6:
0.2370180104882880844922372784516695345167977027602198742986982879894835289891124556365733656671786614
1/5
1:
5
1/5
2:
25/6
1/5
1/2:
6.268653124086036334577413570163990370438121001357565651491404812511749226001770961075660456204288189
1/5
3/2:
4.477609374347168810412438264402850264598657858112546893922432008936963732858407829339757468717348707
1/5
1/3:
7.388519859600066363617100387237269872012940741878028896963130285294862522911242470357738261220429418
1/5
2/3:
5.666577208289452130910171954708807736517710264472868784964506449761311622819401133640304675330435551
1/5
4/3:
4.617824912250041477260687742023293670008087963673768060601956428309289076819526543973586413262768386
1/5
5/3:
4.358905544838040100700132272852929028090546357286822142280389576739470479091847025877157442561873501
1/5
1/4:
8.457040093474584968552562973561784779410916669532337424407714903768033470073844184961674516407382286
1/5
3/4:
5.454145592113897432500873659121652532883653216658858125109584096689089291960056279041757304256162262
1/5
5/4:
4.698355607485880538084757207534324877450509260851298569115397168760018594485468991645374731337434604
1/5
7/4:
4.305904414826761130921742362464462525960778855256993256665461128965070493652676009769808398096970207
1/5
1/5:
9.501501389884367415030576544210529895114850992782658177431167768389650645450023256099648294740771316
1/5
2/5:
6.838085412939917506378997423604562942499114082728552447797917950685725064737739176891648966974209775
1/5
3/5:
5.872250803102905394851606894905352654892264963605387976319599460556259951924960858807830664365755581
1/5
4/5:
5.344796660577975567125921862534413199507254626332622930033316281898105748395374566139019389713483308
1/5
6/5:
4.750750694942183707515288272105264947557425496391329088715583884194825322725011628049824147370385658
1/5
7/5:
4.558723608626611670919331615736375294999409388485701631865278633790483376491826117927765977982806516
1/5
8/5:
4.404188102327179046138705171179014491169198722704040982239699595417194963943720644105872998274316686
1/5
9/5:
4.275837328462380453700737490027530559605803701066098344026653025518484598716299652911215511770786646
1/5
5/6:
5.278007389475821628211924140950690702891630937648301362639677743791130397283818886769180657208027604
1/5
7/6:
4.787597440263337699153635876813446505699503140274967944612273988744620496292496697705424794909601585
1/5
11/6:
4.256457572157920667912842049153782824912605594877662389225546567573492255874047489329984400974215810
2/5
1:
5/2
2/5
2:
25/14
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1/2:
3.679093980405880821727234125103653407749846442767473937598461789766036019230050809216447641489620499
2/5
3/2:
2.043941100225489345404018958390918559861025801537485520888034327647797788461139338453582023049789166
2/5
2/3:
3.110449856159786054921516088619209562216500803088764762796513998698735223635161036387055682561221638
2/5
4/3:
2.163885128148367275594569730485025252469324048775774058863225184041113895813790894028175445750377838
2/5
5/3:
1.944031160099866284325947555387005976385313001930477976747821249186709514771975647741909801600763524
2/5
3/4:
2.913237418670210170507536907198722536453292154795810014406661793546613704258554174582918059675877097
2/5
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2.233649309199920089246229031920752325562603559292908969117088307178129813668086438641113958985349503
2/5
7/4:
1.899937446958832719896219722086123393339103579214658705047822908834748067994709244293207430223398107
2/5
2/5:
4.226169203171729043605006136485895707765385956281164821861049769301343839509846226748527756435113043
2/5
3/5:
3.303265999194124105185898796051820170875600942690217032098779457995221345016496730341382940445813313
2/5
4/5:
2.812606366751803083230257631281275481685948478865818611446337763124706839263086023266498400918911525
2/5
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2.279361804313305835459665807868187647499704694242850815932639316895241688245913058963882988991403258
2/5
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2.113084601585864521802503068242947853882692978140582410930524884650671919754923113374263878217556521
2/5
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1.981959599516474463111539277631092102525360565614130219259267674797132807009898038204829764267487988
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1.875070911167868722153505087520850321123965652577212407630891842083137892842057348844332267279274350
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2.751469311239142509362543295960293686596783304002529428583563491566114760821573054848615685310085951
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2.311655242984993915875189822329723160152302286238159129932775314719510256331877409519068457035293659
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1.859100885972393587407123848621820058511340070271979343637542899706834297852414226249064652236544561
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25/24
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1.261137235674570789935972625574835451592949090597760008684898904006026878244515820076558679904668798
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2.232856219352924131229116280101318240542752379981587162149957788075672871502823692695438140371458354
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1.365846358155865416746052882858796710342953964248388452022214405735028774187822091509909092524405945
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1.175187483869960069067955936895430652917238094727151137973661993724038353422538785629177968616557029
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2.047778827279440130119021761047293952219508545362224650809558095769081387684272327020395462418489563
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1.427444574275983663980957983056757632767016625383646124807873275378204431197189022420586956762236118
3/5
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1.137654904044133405621678756137385529010838080756791472671976719871711882046817959455775256899160868
3/5
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2.415344208002471821259675162453833473393378637463124895781517699160391897945865778313831667280986511
3/5
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1.954054511539097586516128501068068199633339327978389960575457678808266481720022004634465937794295850
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1.468062700775726348712901723726338163723066240901346994079899865139064987981240214701957666091438895
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1.321306399677649642074359518420728068350240377076086812839511783198088538006598692136553176178325325
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1.207672104001235910629837581226916736696689318731562447890758849580195948972932889156915833640493256
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1.116602578022341478009216286324610399790479615987651406043118673604723703840012574076837678739597628
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1.897377986983526389233015796438691319034743150149640075450162568567442634204538267190168962392080180
3/5
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1.496873665048827354749494485557474682004014006468580893522733919736617058740575925604751922324142594
3/5
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1.103126736618329296065706858394587976182990203575372136889629400329908508258452480924516838600046616
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5/4
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25/36
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0.8843414686338345245290247656965111467576597177327534262451579377831279062575681099104421303903926005
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0.9771818491628271566441173488718083857755090344280202575859605752961279098635850264881445405262923831
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0.8091480612966261292728218065287490979986789077839770145561737697133059435225167037244327214585356758
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1.032363413887746483503391750310067200137298374126412010329240971152377107741420369397967036942001551
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0.7766307794247541742397664291398693599215059019836817544072804268756397561844814207207090963017108401
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1.516964232792923191134782512374917155728750692017848148950049113963099486350205696847609308394011615
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1.068959332115595113425184372506882639901450925266524586006663256379621149679074913227803877942696662
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0.9375354555839343610767525437604251605619828262886062038154459210415689464210286744221661336396371752
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0.8374519335167561085069122147434577998428597119907385545323390052035427778800094305576282590546982213
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0.7584821163964615955673912561874585778643753460089240744750245569815497431751028484238046541970058076
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1.463924544872822508762645044903379378815939097598634569066972580118615251054806265998823388171649775
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1.095012346415439945428249009446885515497385946978850324740619317866985729808680098138402164296855122
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0.7469002779963380146748189004609078463346628048972625352382513163870485974769419724483792796794131504
6/5
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25/66
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0.5267775734526080953426397958121000311292538656602996345791096481102310273951068034517361727902763184
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0.6023249885543532361644375315682556960880114735226653992089508384751246621938512883443808365125350069
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0.4670255940897900107892998863770995387239871097093023723871845975078004084741264670582668688459150179
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0.6480490493083973155978975458668034313724840359794894578090202991393129095842026195372930663913702901
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0.4416312220335139621458197294835346180472593697699480263246626798938533839643770266430572715996892520
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0.6786786707060262439307554674436092782224893566273327269593691263135464746750016611499748781957693797
6/5
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0.5698404510783264588649164519670469118749261735607127039831598292238104220614782647409707472478508145
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0.4893542335919087829043005745754460545743554136337823313599666217130216626604134049006525553638129651
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0.4275837328462380453700737490027530559605803701066098344026653025518484598716299652911215511770786646
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0.4186679579171725247127385622118474909750103863814094153336603181219828448400702448521296132105786043
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25/84
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0.4303033895211556516640039912401933810233738529552601096606388058205890080970819659902277943262714034
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0.3762640955031999259985704945910334147842541294058989632415137901651695835042533511758535099872445530
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0.3534767343179223664923199482950927243421588054352853404740135644343717335804110221940851032973763920
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0.4695743559079698937338895707206550786405984395867960913178944188112604266122051363053919729372347825
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0.3963919199032948926223078555262184205050721131228260438518535349594265614019796076409659528534975976
7/5
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0.3409219838487034040279100159128818765679937550140386195692530621969341623349195179716967758689589728
7/5
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0.3329732930099809410281415848277886671960609081084142108007539521862986802123726972386384451766945483
8/5
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25/104
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0.3110790398479306065179883362370257610663277309571870659342046453975395641412602667841941681632062723
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0.2904650818836085290948967036946516244282990844485425036609302263502243103098258619887085762295729876
8/5
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0.3293651192730643392626829766982500190990970869267897585156615044309625315380726061337043182655890697
8/5
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0.2791506445055853695023040715811525999476199039969128515107796684011809259600031435192094196848994071
8/5
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0.2720038528647935250572975815219531996067647077309136775892236877525801801185225295430315492438471108
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25/126
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0.2333791127373727986361203865192180239582693372335150998384690944558614594384931841304014320606171716
9/5
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0.2269064135685077512936158684944530166079988268042316562749117923201160296132481941615329457253913368
1/6
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6
1/6
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36/7
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7.285951943662744835459825069342793745665104534624727823249888256416217062085307523364018181266506332
1/6
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5.464463957747058626594868802007095309248828400968545867437416192312162796563980642523013635949879749
1/6
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8.413092631952725567050114474301764812778133232543916577091963922457708595890819776425113598910148708
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6.677476047133832307371998174110188264954081255961865310459529794878861978254129961912953476675550199
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5.608728421301817044700076316201176541852088821695944384727975948305139063927213184283409065940099139
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5.341980837707065845897598539288150611963265004769492248367623835903089582603303969530362781340440160
1/6
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9.485663553299684607559853900678869197380739897071152750291410649885627782054041566025021286911386026
1/6
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6.462108662147073299537430461712209710252485819038916131953636638237896396517850554923729744898689932
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5.691398131979810764535912340407321518428443938242691650174846389931376669232424939615012772146831616
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5.287179814483969063257897650491807944752033851940931380689339067649187960787332272210324336735291763
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10.53271436857934293813799892898958231253890690860492947814700277523816509184349273495193454880112349
1/6
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7.859627826148979313975645395921058744517827773209741041771436070046334871528383192364832753919998440
1/6
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6.885618859224605831847674633564383537218464429755472110204576030788438470206649257781858842691055932
1/6
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6.351071609209551683483844254791935989884838492477331883495592043628517232890344569202732552921759710
1/6
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5.745116928316005238984363052176135806839403768329961533534728786493544595550996037246509753891521902
1/6
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5.547972583163985398100455573591335584365525486971581911838660755326824615196505782845764296884704781
1/6
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5.388745194175778477098180148006908855214450423286891216681842111051821411466073332177106920366913338
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5.256059262794111738055595245345050474387452545498481558754972725761531503081664471064330388624904587
1/6
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11.56572777995998878627418672995763835752281937142112850237168264238230120433900638209343481907842178
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6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136
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5.782863889979994393137093364978819178761409685710564251185841321191150602169503191046717409539210892
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5.235987755982988730771072305465838140328615665625176368291574320513027343810348331046724708903528447
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6/5
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36/55
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0.8401884752499601647414047406257139259406534891404245051544127742676457737829514987151863066443966903
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0.9313010834451029262046770776581941105237656464428976953526844041721737491690991496791750750324452186
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0.7665509805125305288751155630018404852319580784712300966978529011929088872758650223527834927065540820
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0.9855439328251063657689507074455088854988338259444369128551233690986718098808952516713130534873949346
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0.7347558325136930619259355632001755570020728785566816103259646927673989398662486124733774335682538502
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1.021549817317900960299082091796907877979025342770638973414131176217638141409771397439196256233811794
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0.8923684252667489219554194473384736280854432337305500849460205918592804629691588285995510330735413895
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0.7942512503651970931673089380441033428517529465742679385605331682375341259460857862656521237920335635
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0.7170242668764844940878261444424715324812762927013720338287212637315666535778642935504441084922366244
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1.411428194462003290422763371151353568463190011166886991381544611771078352223091069870150184163515458
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1.047197551196597746154214461093167628065723133125035273658314864102605468762069666209344941780705689
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0.7057140972310016452113816855756767842315950055834434956907723058855391761115455349350750920817577291
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36/91
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0.5464463957747058626594868802007095309248828400968545867437416192312162796563980642523013635949879749
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0.6231920468113130049666751462445751713168987579662160427475528831450154515474681315870454517711221265
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0.4856346216097332587179635035716500556330240913426811134879658032639172347821185426845784346673127418
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0.6695762508211542075924602753420378256974639927344343117852760458742796081449911693664720908408037195
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0.4597547664768668750659041435210267778045246827774722939729860058825380835467245454095934205856775446
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0.7006240156482933218273613478263580252243175327231660406749669251821395848232921996642085065721368173
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0.5902098492727644040532399546373761259963324986139980757275171016305132569357984875367834358387983810
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0.5083721881297904223677528441515951750202311720081972845926266142501718312703842766204817849402748432
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0.4454287510842467574623385801139873283379197072456340304029637903187738561933613958529093549682122532
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0.7228579862474992991421366706223523973451762107138205313982301651488938252711878988808396761924013615
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0.4363323129985823942309226921221531783607179721354313640242978600427522786508623609205603924086273706
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36/187
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11/6:
0.2205356553846880141285567767423989950723734392448260924033663455892309925348579796672109662755492904
Definition
Euler's Beta function is $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$ [1]. The parameters $a$ and $b$ are positive rationals with $a\leq b$.
Parameters
$a$
—   first parameter ($a>0$)
$b$
—   second parameter ($b\geq a$)
Formulas
(1)
$B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$.
(2)
For $a>0$ and $b>0$, $B(a,b)=\int_0^1 t^{a-1}(1-t)^{b-1}\,\mathrm{d}t$.
(3)
For $a>0$ and $b>0$, $\int_0^{\pi/2}\sin^{2a-1}(\theta)\cos^{2b-1}(\theta)\,\mathrm{d}\theta =B(a,b)/2$.
(4)
For $0<a<1$, $B(a,1-a)=\pi/\sin(\pi a)$.
(5)
$B(a+1,b)=aB(a,b)/(a+b)$, and consequently $B(a,b)=B(a,b+1)+B(a+1,b)$.
(6)
For positive integers $m$ and $n$, $B(m,n)=(m-1)!(n-1)!/(m+n-1)!=1/(n\binom{m+n-1}{n})$.
(7)
$B(1/4,1/2)=2\sqrt{2}\,K(1/2)$, where $K(m)$ is the complete elliptic integral of the first kind.
Comments
(8)
The Beta function is symmetric in its two parameters. The table lists only rows with $a\leq b$; a value with $a>b$ is read by interchanging the two parameters.
(9)
The meromorphic continuation is given by the same Gamma quotient. Its pole hyperplanes are $a=0,-1,-2,\ldots$ and $b=0,-1,-2,\ldots$.
(10)
The recurrence in (5) reduces positive rational parameters greater than $2$ to values with a smaller parameter, multiplied by rational factors.
(11)
When $a$ and $b$ are rational, the integral in (2) is a period. The value $B(1/3,1/3)$ is the real period of the Dixonian elliptic functions [2], and $B(1/4,1/2)=2\varpi$ is twice the lemniscate constant $\varpi$ [4].
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

a = QQ(1) / QQ(3)
b = QQ(1) / QQ(3)
RBF = RealBallField(numberdb.bits(100))
RBF(a).beta(RBF(b))       # B(1/3, 1/3)
Links
Similar tables
Values of the Gamma function at rational numbers —   the Beta function is a quotient of three Gamma values by (1)
Values of the arithmetic-geometric mean —   the lemniscate value $B(1/4,1/2)$ is $2\pi/\mathrm{AGM}(1,\sqrt{2})$
Complete elliptic integral of the first kind $K(m)$ —   the lemniscate value $B(1/4,1/2)$ is related to $K(1/2)$ by (7)
Rational multiples of pi —   reflection values such as $B(1/2,1/2)$ and $B(1/2,3/2)$ are rational multiples of $\pi$
Pi —   the reflection formula in (4) includes $\pi$
Data properties
Entries are of type: real number
How well the digits are known: proven
How they were obtained:

Each value with neither parameter a positive integer was computed as a Sage real ball with arb at numberdb.bits(digits, losing=96) bits by evaluating the complete Beta function at the exact rational parameters. Values with a positive-integer parameter were returned as exact rationals using the shift recurrence in (5).

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All entries were checked against the Gamma quotient in (1), the reflection values in (4), the three-term recurrence in (5) wherever all three rows are listed, and the decimal prefixes in [2], [3], [4] and [5].

Table is complete: no (it holds every entry with $a,b\in(0,2]$, $a\leq b$, and both parameters in lowest terms with denominator at most $6$)


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