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Values of Ramanujan's signature hypergeometric functions $F_r(x)$
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Numbers
$r$
$x$ 
$F_r(x)$
2
1/2:
1.180340599016096226045337940558488587233716634881447299515864399404304180720715794978458616195807954
comment: A Gauss second-summation value, equal to $2/B((r+1)/(2r),(2r-1)/(2r))$.
2
1/3:
1.103845772797212254324837694207631411924760802239391086997330526905118375631119575423168443314817465
2
2/3:
1.291675482135082644544318504760649923199863516052326580950539239585172786733440152066839813967604411
2
1/4:
1.073182007149364375052841707970349795669526818631368625787232257139788343028625222768722717743272285
2
3/4:
1.372880500618350164697637575007806058094538625348940055855540038870350633240892464904292347403572711
2
1/5:
1.056549197563300524078647664524631546162020827872518214905496804039471255692357951593748576322171397
2
2/5:
1.131603977657727948718251611721629596963366772407109106123290929200991062332448898357341008018769227
2
3/5:
1.241133377096689973536682961615741508733165956734589222952496929906656479540528590897435022372835882
2
4/5:
1.436981541347583904374530510658727657178127808659478463032740900956857996992894943971118927044229213
2
1/6:
1.046091783138450905722887670068762559303746574800383734428214108534602441219885573271726413888274227
2
5/6:
1.490006710996101632048041757023586658936775098622123546456694903920849499466878439839757245006245228
2
1/7:
1.038904133044564197904811634148864850543294789667688054067546601491881507649076613798838908792871211
2
2/7:
1.085836219042230225420359741943770555185613239379711403618484105693628970552675777371878731716489871
2
3/7:
1.144560363318554196151258236386538674584153252036171984443250885612698680823735943280588129450616996
2
4/7:
1.222207887117855933150549202510695161112895220266936745908836236619192022096820532314266970327331576
2
5/7:
1.334976602403716770127827025536283176651684589423227647458068504567674708882393623447928128047799559
2
6/7:
1.535256663440011474732454706314617512498973195528800109170420969834694307748934528735174488793888015
2
1/8:
1.033658320257222080650660793407819301371959086449491349266378925511332812850157652680195437582610948
2
3/8:
1.120812916187724835510613192799134646059099550620059458865703824388030737774532390034712889934777390
2
5/8:
1.258938040853862956996168399328334822946195216227027907564484731201390225011645978979010358952157225
2
7/8:
1.574740233062900749709511580466414225965172203836240246998315913582919707616074949971864542148614141
2
1/9:
1.029660375463762623523228181255854543570903969011705041891655029152762974930669348678219260552679533
2
2/9:
1.063794415811677049552541647164834182538425445440191871843236008928428344208836477382503166338992521
2
4/9:
1.152070109037683005025037166134955830569733070726355371852509034427494430837553305319982903163462641
2
5/9:
1.212277737388446126114849206956631294516399256257202991733337430972124890082201154989536831079765176
2
7/9:
1.406601154881691402593472192886194359659700508109655428310351240159337356982388830048124345702344339
2
8/9:
1.609773010724046562579262561955524693504290227947052938680848385709682514542937834153084076614908428
2
1/10:
1.026512044378341921684211979827646737352861418956689804570238131348948968554188533262148256466594475
2
3/10:
1.091095910362781566395412210019548535491316580086392618691450030536134538831405356974682221464949521
2
7/10:
1.321217206769961628317271699220734673082433136489992485826932636346228350095608636981900391995144021
2
9/10:
1.641264414342370733286999747367830456947182410886698750765421537225059699558810851709768888498397108
2
1/11:
1.023968347371686311918078592385397263566137880870782763162684166767298617336501277303748294085862560
2
2/11:
1.050786193652262400764871862066183695695740265705893064553867867085500756023258054733824801839352223
2
3/11:
1.081154987756952037335394327896449304141077364425785780428448731523391831955526952099289538665511279
2
4/11:
1.116064294972751678334036987184019823108578304092302884332444163788414765343825632784028634390621069
2
5/11:
1.156972843564388095923427149322346130391954965506711732553570552060545352478125530933061521820453676
2
6/11:
1.206157630886530125818869972389303717538294388102229398265886485654936308993286211834681097284875567
2
7/11:
1.267460899895319968535438611378616672501836157799738102768656409968900058388790192092353843626673639
2
8/11:
1.348141232842108551990977939875021899707862668809440574244776569347689319201985389574440550016207674
2
9/11:
1.464631479355372972726029531108621642886383902276560476865346851375978385301147961042338820165628426
2
10/11:
1.669869780325460827882248226975556605010401248986942310359903568133620024908545856235818125462502544
2
1/12:
1.021870277329714933871211801472596079114135093580612745466398921978410563024149678317763851492740071
2
5/12:
1.139077066794801910321708704362096045519657973677261905039448176563141262831351210724591951014013081
2
7/12:
1.229921800246286962902709752771617769830256219313066368725015860026194850024480467362460578888731842
2
11/12:
1.696077212872532954693271801515080179397678020550014541069491364768418041560008714483898940035600277
2
1/13:
1.020110111882724721261218964732954350123425440956908760486690138965610284987851579196663536369358260
2
2/13:
1.042193422634593581173941728438318205007332912682223306261470608447909946955838625252558996635367971
2
3/13:
1.066642645093745941036586777409462423736337611124628262431136681264199707445192615691394090248044245
2
4/13:
1.093977585650621866043767772047057785170375733254121698634665587568703407282034434420244383965152342
2
5/13:
1.124906021881407111376881049044040164593873497193532109851692742489475738877758467797506643802871598
2
6/13:
1.160425804683368851529789391295459428245010154954873859861313348443223420256440643195194385286232406
2
7/13:
1.202006862033232502106623400685798777711102620460344754682003328650427024118618573622292149462961127
2
8/13:
1.251941966510046157036632596535264582667013365900994138398781342252469091615815143943269697699483548
2
9/13:
1.314096845887786291793009468625821140463627976372039311114044886005472434523175274910756138084193732
2
10/13:
1.395767916162076667984583505443053047532909109688287518964830459824024590711297160623874868922768697
2
11/13:
1.513456980055890831709486630471705070798368350152550220009044787260415079096394982864461581718456572
2
12/13:
1.720260173452507520673349768053572467195382594133667874402303970715435975182446033208068545362434867
2
1/14:
1.018612272442055661737296385398174558244270099180313059638775444407589491241279895906221463560697468
2
3/14:
1.061180663199903468837852110355865556975849791487640724897533585311251606645413831885137123866916270
2
5/14:
1.113392656565712370997064080743595105201661152885422023779003499560355492825776312331350480049379061
2
9/14:
1.272461453890228237715142827999164425920664388076998163586392343981316239943484043169653901884645834
2
11/14:
1.417065113963005640602884478638049908496841012485567388317537635621029109310292856766683857881269826
2
13/14:
1.742710866006256297904148690524989786144267935687524599296098590256030695265110854944903932966417713
2
1/15:
1.017322163244967963750161950546178246657604196859818594165460987822604004040305643722842590097950505
2
2/15:
1.036091320472303650089603604880866151381197315781013151400053945278407989063476332847687052929146234
2
4/15:
1.079002073958565572052906690825655917669994085919183566421018424799141948511349361335268395884031101
2
7/15:
1.162989400386374086724723272101540007368323166988201242875520399085697958155044159364751064659706630
2
8/15:
1.199006332193945816492412836051988290953671150104481152379223059357967724876315745958503456411792251
2
11/15:
1.354516421872935054502103315695128584639322620069382861441601808494640682098806461199029483200989610
2
13/15:
1.555625054039474270764282971071874899024793635427561149121165379715885447919195083681140968150244092
2
14/15:
1.763662272425612825015627157523997331881953816798130553323621032498697905788660652699087137076933613
2
1/16:
1.016199360097058232069473923609701162536348851915089087262799285576704585360569233724687760164709106
2
3/16:
1.052571662629471640880865951563740731664628671070578929026850832771926240313808530943832835321326036
2
5/16:
1.095796711117877009426349004947810831796700629463077706926102415305679178080502821117403759068185639
2
7/16:
1.148755946759723961801792420359568135065756009745320982741513966794933693091897570740623877479826685
2
9/16:
1.216573879219003681904988045993711789197304367274699467816178145163008374639119574774470163177755195
2
11/16:
1.309743139363102747792102910659417606371858468579685397979577698360598679909269286171733142810977005
2
13/16:
1.455687535452987105533868557863364774384170551695220313629073706570678389882645239045175230146262371
2
15/16:
1.783303179974245991160007538669181117318624940708582272198469865556835975803874930841290312339299434
2
1/17:
1.015213291235736088422609798030457119201013759708114646857042057103150487523123693616712283541499546
2
2/17:
1.031532986712177127253749530412711561485410247653018564802574927094416765536702939377222694798002798
2
3/17:
1.049118344318675317480862125199109025364348756781479926897587577423980165990418678519252625307856946
2
4/17:
1.068164826944553590611884119062111818354514259936040120869954375364476371503878074694075034910262844
2
5/17:
1.088915864350570993432780574612118921302325646694076597701671158316400225827498810635723742806910167
2
6/17:
1.111679789305732201868325345575801968641698567187440023585875055262483957799805863716148684964887497
2
7/17:
1.136854935614305570532615031352238661965235551724487045174296334716506511766709438384085465845741398
2
8/17:
1.164968109764743853915255582562841854727800572772819740548181608602093468387729182730513402677635738
2
9/17:
1.196735988596297157342510574736695669469507652169726518550601269194219556060769678596525068511118298
2
10/17:
1.233168025219457904313450609457345555130964443992508460910896332086054530109818816344532624880169049
2
11/17:
1.275749757812629851251275193753069184011222979604860368194593187595736885306583945404208096730611761
2
12/17:
1.326795832988675034739892406239936001848318943385655154816840101649691842554989983466614064299442579
2
13/17:
1.390204352316873422137394001003122358720363549517331529869377830981105872141488863880027885962302218
2
14/17:
1.473323532886835135694023168611722060535292465920496068968114673340302530632436589054318337898616999
2
15/17:
1.592748823190620769577493121237489741796410938579246285889247041285337384431899627338978381497656780
2
16/17:
1.801788718747199678160073007873291296435212563055638801037967020457484811686834217732303485953389945
2
1/18:
1.014340407427455744278654621278197332516084213753450676650343792674372358104693129741078896567330302
2
5/18:
1.082964356660988557342229012771088257341532796520974369626277665900261934219951681772913494828366804
2
7/18:
1.126747884439972355589626782749870871451110982273049019550826921585642925051987979343676521333358921
2
11/18:
1.248893194668392380527964863909245978827246644444190483660904095980319636789635691445296033123351366
2
13/18:
1.342943396497485001605082034201356067783350605315464574726943345207226625154995166253019337542230116
2
17/18:
1.819247930243456894386869446225929504956215281300846213649424183995762799054391386368117235900224299
2
1/19:
1.013562274667718193946377850767723061177626047606004976550570207536662451751925474621009660683034887
2
2/19:
1.027997901050387243486048904988958877659539963643110492206109615428238841154562156978372089792286595
2
3/19:
1.043417316231362148894732448045334695024176942781149390171863109925404493659119675310412657233932354
2
4/19:
1.059952814282888052693406819786854022096662160276669545701538960551452607333318934595451699355183990
2
5/19:
1.077764663736496486953739001186621767989348401425820445239524942825980428916984748778104305424737883
2
6/19:
1.097049515583030437312617374397593711683237397747321491358115265141800090947350247036756539815693137
2
7/19:
1.118052211425971054069163600509312892625540208417859539549185581650078986335846341418763176491182873
2
8/19:
1.141082782668507737259460753590327193202715571230277399634784762124751489955204931714302865970248652
2
9/19:
1.166541636474413242937964981411311318265274654904536005068149963105360934694831425391000844174572165
2
10/19:
1.194958153474785130703940504454474453727020780735600864689572257469129010400854946314325137076159034
2
11/19:
1.227052272152879869365189745023761282515882945237243197646508747531811318959208220049687764959968673
2
12/19:
1.263837682501684789509316834438242733889511861784249355232524403570169605789461680892878182443963847
2
13/19:
1.306805600493050096442122172940450118006914277531529554954629156269871684355211258448573362154049970
2
14/19:
1.358278593583469060753262773071273445310885491189421576421826758252268811307595142156036364366442004
2
15/19:
1.422166427708889615303218270895779638085660622492518445794346437564964549938147559570258869292547978
2
16/19:
1.505835876781865341815812464102119104472518662105012762725044538856008067215423127100514584162501864
2
17/19:
1.625915707366210368492636222567923160493809611608175284948734765205566393619755066190731956394041059
2
18/19:
1.835789318482390737476802696936049683261710547847253469028191403664053428418486703424781943449399342
2
1/20:
1.012864256588329839782826986305014051078928263637837778507181187045743425613137891710257169212433903
2
3/20:
1.041036768656831824337554910102229192405652702156200310209040796469052381265628645436735637623989499
2
7/20:
1.110488080135024470558511455026001465080906251943916205890561460753498943675909565996798583661549540
2
9/20:
1.154754378957639790044855173713824551480466482988060233564316471798294060325679291469065439429278075
2
11/20:
1.208893144120206232516306408908837768180273166082994846268712718083353972297833321527288318451408631
2
13/20:
1.278076835410447316078793692074485929087733337978697251151369571802296811486588625027312326239558693
2
17/20:
1.520895131706989785265585357180785292577555282365086745968252303894393407419229800951214530403499361
2
19/20:
1.851504997072928624504447168973920255102443321185458361659457162426346798503144595800969965361656560
3
1/2:
1.159595266963928365769992051570020881945165263439782855263105059747973757205286258658085251051146071
comment: A Gauss second-summation value, equal to $2/B((r+1)/(2r),(2r-1)/(2r))$.
3
1/3:
1.092060148081664394316757805253605515050933694436801793669960490099214146997645089823341657245948803
3
2/3:
1.257546863411625467745864646361554193208060216425676017110925090768116861297340574645820318884594719
3
1/4:
1.064925231128516498437819437604916718248282322874928859475392443772515623080612622146534197451821389
3
3/4:
1.328778200157056363878073355047905281137317875982473604891262407713582251790491637816553512224526597
3
1/5:
1.050190162034047216423273458755210265735253912697437725038968163649388321021812855122591150688006205
3
2/5:
1.116591847517491603078398227585544469068273798398588049562491365127247729007411521979714563385367573
3
3/5:
1.213125929265749290045665105498391869491955138301218750013311846419003653696516756626615190713296483
3
4/5:
1.384900768403713295980836955613618765996932275485885300036548344312948779899235384765120985000191027
3
1/6:
1.040919744138056218742701069892397123860878861236686382807648423603974286719266533999771862365356725
3
5/6:
1.431264829472441829703947649619931413395224683882458059073668339881763624192223392310901044689123357
3
1/7:
1.034545122849013258464320235923431824838519691492821668016344638509330279300214298221229219460208758
3
2/7:
1.076127821155113612750818858776430927664391876945783155757827903850477326425261991200767457088468650
3
3/7:
1.128032332151065543728238163703959988069884239240462769722960859006034030405693314087008682285193629
3
4/7:
1.196473564871515518228785425183390940789314777619902920985397928655142565693070797494023465587804910
3
5/7:
1.295549635583874100749921062420089511310885550300578971974390063341866749753562644478408989657098971
3
6/7:
1.470791296583444910638740268211626314580220190451585807085004608970441862434825710374791292919191265
3
1/8:
1.029891211707070937723394681657891254219688227846421021674796234582582177220985558207517716686222714
3
3/8:
1.107058591834453422500498838406624759570443201432808670785993628303478823787075589877961046097012653
3
5/8:
1.228782449126488018539047223851035872614005142163756355285504314300773493210125954143540802655757776
3
7/8:
1.505254038857066011299767910248401498208259856931645808635186143997248989499343595330290706097970142
3
1/9:
1.026343520841566885711818335651046428274180826014178666730958109479707060905841596720020526615780503
3
2/9:
1.056610187209305634509752575084449105338086804837491860759765218539835253267529709258419364649947724
3
4/9:
1.134660648114355403631337399466476138628544423535191018086134518681213796429802162682889092477494993
3
5/9:
1.187731753381443821101320706625882493021599661502139327862103330238600432166839150503617694239926203
3
7/9:
1.358312435705690357986949710962349877080998310068052564784840695120381752100133197132806489600122829
3
8/9:
1.535812809958194137289311853725147146475663454326775827621496728753103561572492083134212409167445065
3
1/10:
1.023549238829423071188169685444329193566581063931538749518423631994710790474632618839510169768428974
3
3/10:
1.080782209186556170155428610341809305181042107791418949037651214294649120395376910063142833674184175
3
7/10:
1.283478964116754298216757307564906253314665492204852243043222340743204179162304530348183365772283111
3
9/10:
1.563268212972069953487391346705679048399059802492935756214049227368641731628039675849038738599696945
3
1/11:
1.021291258951993964175132732967530845368701631760419883631073286468895655282506451238914602282882422
3
2/11:
1.045081900472368174757478699398120070907417237788141519093612942437713213471904111971556078089367493
3
3/11:
1.071984369175617240301058194420032415519161751845227138677651698438176623395971863108959965423345757
3
4/11:
1.102862073496377898253471085285026854232749670271011361476607297593240578379269336624743819136117627
3
5/11:
1.138986857686387145575441200021526791554401695152048963342824066811603058194723902345884560135293771
3
6/11:
1.182342505093989500817360735037737114605461728696797534653525868456243993388825683848959597136090752
3
7/11:
1.236273815734982676595844207223051262747844837511365292545957883638714865668787720970121533776357960
3
8/11:
1.307094249651611530094248076719487748542299165500513150290411651094082545805765175936547745881581498
3
9/11:
1.409083755732918257279562196466449984646026627172505421412440893913742836941909170984063878340167986
3
10/11:
1.588196520747996037425192428266952001413766176267406643122997986911221632162574780003881023863631072
3
1/12:
1.019428623564650582115552241555020263029813186337083625699923352499207753720690503416460063770942168
3
5/12:
1.123191332955536330762430668274561754731048153790780541934720918755487015510744462621572554139964393
3
7/12:
1.203262266949699656021101941512448070911519524203632900723997819462483201422166411948463904318356582
3
11/12:
1.611026540492948588020491489165422642948122643896190927713546621564308095435682632975074176095522827
3
1/13:
1.017865815072909931233943626040309177904759727252698327763370155671316535572066881017700059335175674
3
2/13:
1.037462634017176756615497488345219493947877967939783041876786930083652363632701013616736121620183857
3
3/13:
1.059133398306179490824202108475302721907471182301008626986835885229666096254617498373568300318841640
3
4/13:
1.083331775453141956548798651000516324767649735588452863624119467166554584390100154171412570148358356
3
5/13:
1.110675120737380195582432667466104838982022893102653969305234230188436289305290197253675535460785031
3
6/13:
1.142033268242232241904824490034209385141866052996097430403510750045783461942162776833103312659598595
3
7/13:
1.178686742794826341781312204544414686089031794348372126855172197482924115842973041461090811738117608
3
8/13:
1.222631547105759605999748721490235032170250072241761285772661408306138390468082214314833726061878652
3
9/13:
1.277230660165248523624516185905433691627482058146028235747823129006592663488515566209189968288370329
3
10/13:
1.348826806568550213782767521304014438951203287400827667687017345611841437104871349038266636715121629
3
11/13:
1.451753226491734885413015226515318002565711808994718081527379666581173164549386453142002032706613611
3
12/13:
1.632086095029062496193992847374845067830068229096750639901288741749026331465324404170393021123415646
3
1/14:
1.016535804257895773654358007444890988082361277312852089936939469106844219019574072056348408979585150
3
3/14:
1.054294389471062945170697939922529296869571267432388386228869934392316911228865026276832081458387311
3
5/14:
1.100500675622999564520039247575022232318303375162606518432695930114725310817632520566213762565329908
3
9/14:
1.240668225164307370000221494546283062153991382258112239008819623990529783304894085084675161849366503
3
11/14:
1.367472401251472832319075900220296509866029992241945587231126002162514111456154510376255760332130404
3
13/14:
1.651631496535268959487989820016879410732581009005858006649553693783820955788854773777888498307678223
3
1/15:
1.015390162869452254003286088442590970957218679943443223976819715009351269604963173684750254628884664
3
2/15:
1.032049844898228447681951946533144945509187042440665424241805674742406475641386744422496386782429043
3
4/15:
1.070078477982563980030700669835734036600392660541959639328528923231932448518532442973202778798868426
3
7/15:
1.144294757438132997198589163071957453934388850632375763756720237153936521226369096121159614496719663
3
8/15:
1.176043703026174230337971709081424801595153951282712861872327557186188587430087096926418554861318063
3
11/15:
1.312683446319700735042845659053906130893211394152161351085675234172233422715662766959645160444124038
3
13/15:
1.488572559350426622331635895820691056107507568596255786926374545395870940432300191430535294280697481
3
14/15:
1.669866980944424833657833255953531498140640822417128094475176023161235212388373555445744969418474361
3
1/16:
1.014393028031851270198501250229213360691294998310550141884346040935502979543174751160238989022037019
3
3/16:
1.046664676068237687005329495710600415416301125811055512091540896927562763501802197241064075530291343
3
5/16:
1.084941077019895092555359660933979813792603602126112902862538140615751708412558794461028470804510234
3
7/16:
1.131735722976506461417105595079541339749345275988699476595787332817858643991654395257992422993097433
3
9/16:
1.191514152156682500233295445739684633723801136580441190344289502530201640212284809239922532034442156
3
11/16:
1.273409538161852037771099978891342320858588349046536319734746937111849010797239706980237367698114259
3
13/16:
1.401262851064279070844694196322975931415118269605778250673621304675712578660472086475916760272193149
3
15/16:
1.686957949907420765228511934215866246397383048832596619359767787506739098660187509111373296629357986
3
1/17:
1.013517274125744164204829919417776290301568431501880274028734682256925528250498618422714986820844182
3
2/17:
1.028005327775855093209434394808694063798043000382066385729024186878866652195353555551451060302449270
3
3/17:
1.043603264628193551650426518211639665176830129172059568358541616622747505500248695996103276614637009
3
4/17:
1.060481738218176691641716410336454911922607320397469685440036124955070001132929080130861281660833012
3
5/17:
1.078853187035423238907009383483913286206121568517657691404126301378071768741501326971102974895669512
3
6/17:
1.098986567948413946146213149545147817978705351700542074209791368677387777529860452431427985112593065
3
7/17:
1.121229183513491890573442641025621493335958100506292388523177270362666149208462278190494893636676023
3
8/17:
1.146040129954513831689788847521360240419359806771452016652866426021431775844983476355092767825781042
3
9/17:
1.174043660918353839661000778053731911296260098388236560858898656552362142578706598788861831531012973
3
10/17:
1.206118599706407507330501347901369549831287095076860909369714027498320543316468281143892560423599110
3
11/17:
1.243557559455093881284202380183674703886805031756226151460686489301140229759705528838736250233050719
3
12/17:
1.288373472469284428446871566305307291209145175641385664099417908784109110557571687390785739074682376
3
13/17:
1.343954358812194386425690249330078372316718722911492837635708113090822278749712903728871459506112497
3
14/17:
1.416682985041178539651498204281811298055650594514579451097495988522821242080170391783328777490358275
3
15/17:
1.520964926145018587060860363114097060893815036240999865697291470740329306215425776791324159111832181
3
16/17:
1.703040249035677460158443131401376189704015368632791455687916692854588657696654413342264440358701061
3
1/18:
1.012742004122493405094936210954076111637475584035209773410361951147551623477854554728861867679484430
3
5/18:
1.073585985005929233145765221175328987742412954478331427935587193730992168397016979939222006417247886
3
7/18:
1.112302321567241848733994451049559137502195816677575045853826735809883938296291579485976576369523326
3
11/18:
1.219950647524886389512586759289031992138504820038319818675928552721716319752948322397294782089328133
3
13/18:
1.302536543488751620534790620721860205518340727033319896611051102769723527288842440765577278288673258
3
17/18:
1.718226827388671173001126055545214464734270334955332804983986617630859377370306159126557040281124282
3
1/19:
1.012050858509383296798168311602721405600175343474972280609850539887725334690779966228985958240320030
3
2/19:
1.024868059188218484114990980582776971953694230653642174241304445399116576358618963361233950238466651
3
3/19:
1.038548070096717136485567395427990399477671386949221743037677101705734966544909460115622640995088132
3
4/19:
1.053206407460095294928182461915763773036530683592168030299700917037957715955956559966374202568598841
3
5/19:
1.068982959494198050033317325573434344203944714878696020577435751719549568826403396242315756415896954
3
6/19:
1.086049301715919700873298478272265869015068224960312846715484286241064245009797713570600500257169872
3
7/19:
1.104618967835855852829911351657196570435121823431809949980921895363252315982821698288594977479923407
3
8/19:
1.124962231936159176577645728052333659609485668539322865269421385549575011319110556085096193196004397
3
9/19:
1.147428003756447031367080648942763225536356340991671747382121624344302313398934264973722653211646046
3
10/19:
1.172477374096219486527841481799193014464682739990008506090857278300652421387879232661210406048774507
3
11/19:
1.200737122834585329220022581923403990363528104424418070987994505545291840872801020920214521483501820
3
12/19:
1.233089353413155853668628836306954588001072085387441785890080281458416458974329754633131626855558034
3
13/19:
1.270831072600650621324300550425282888089854175523021270694953873513379894739422457441451711439869811
3
14/19:
1.315981339031860157747820093581762330365226499322578345350469234191791410252644927396041313375225506
3
15/19:
1.371937189696991617159061432242720643541272888919839915409454386099403981530428570865071488197716536
3
16/19:
1.445095727826777250610612347443330366284799303326824592179653731458215836084646645913501464255758537
3
17/19:
1.549888252659733588918352786647744141871417859895164216693635466551730822596419633315127797645843118
3
18/19:
1.732612618754347339100010432519224181603843834137199599056094738825084389475711064906843441959150894
3
1/20:
1.011430846896399243640486184188761295228226850483935991485966150673213653355342400363973477360967006
3
3/20:
1.036436768840294808843795225687195619453818544766741383967300981486094614434888997123379392613585411
3
7/20:
1.097933076595116867677920653218324945692050500336941282626965624999874935524165165027229424829801917
3
9/20:
1.137029373243031782287418947980166112696049897629836352725663496123549230406037829455265658119365125
3
11/20:
1.184751491399005991938464830189911325712169473094346201589788757023955694906881486384213102500834906
3
13/20:
1.245602113862525144571721701417204989232621070999139319312608443305409659507776692298529286527362059
3
17/20:
1.458249973741845631469921584885728930382222493743938446775256764021776926260972470350864553272565088
3
19/20:
1.746278186220003675115938355491152316755529013253150593740976874905771172753221706299820981826218055
4
1/2:
1.133915559726082732440156510347438791344295356805403611236317061400497913023777214458207113170347034
comment: A Gauss second-summation value, equal to $2/B((r+1)/(2r),(2r-1)/(2r))$.
4
1/3:
1.077415715750701203246257098513742012348092021182187093026426589300763704600445579059592620512413547
4
2/3:
1.215502554868454386941982217997043941288277531484048915816265650173538426379192234806255033882957850
4
1/4:
1.054648614831467047906718149577441840630273364763410414893082861937658163667864603886693740230931228
4
3/4:
1.274611308136601952102809952021133623397842247250675660982363975799881719700368808191505969508039952
4
1/5:
1.042268132787808640280111359191757076616454207746347141456642938948577695890045478200055253124276725
4
2/5:
1.097964954833625034644635115029838609229817266028104295629106404832222814579922722381761885136262909
4
3/5:
1.178550802124159000837858875238875243852949017127044523005083344491627440708876190233450341698228100
4
4/5:
1.321072304137970981489625469511684229549308657735090684183580728605552210350786647000383877823149081
4
1/6:
1.034472595975224665565210956965449694286788213307133302690146790981876687220797791144859155481474093
4
5/6:
1.359390804098473584376788492257887972976230618168990794668557788882956618905982520170923667645410207
4
1/7:
1.029109172003885640883409790532973787966389722225236362612312609792303595339076362320466658371673082
4
2/7:
1.064052855706798065059318519246029202243427610149994583873678689162568557377211969188254443666791935
4
3/7:
1.107537793322367653266005014370961059569233422517096275050673312685427899341636698533315407572250209
4
4/7:
1.164678584288915160597719744440372368619041696755789177096439287888089589762692865574694327714695891
4
5/7:
1.247058654519720467529444007209479917722934199631468498536085139193313512364032380250987836581499310
4
6/7:
1.392017425082818827052250945693554995149461510680459772868971988838488279020453270175854537750954101
4
1/8:
1.025191948147477204555870898040341068871121508224311888461749724469665489111473743828715077692710259
4
3/8:
1.089982996436349562230291086266379866968836215243603630021467036634028291166639603728674550705978222
4
5/8:
1.191583327892024549438019833834561300225000412141646104404464790759077035010424432025608253740841480
4
7/8:
1.420436405603149035164429103674611486604556948768265300039881213202036094679225782024480005713671310
4
1/9:
1.022204943866080919860161806857062989134020137789994880270437701856939901361339037391092749731504665
4
2/9:
1.047663807509503253763962332160564725355600096306203016834789709390477727619762834985644836709259122
4
4/9:
1.113081115689331418869325768112246824537379728332091716607837515961784678747445039488847502084758853
4
5/9:
1.157391657238803968433859244888091901309236348718160614407357508950030616405857074098800775120177135
4
7/9:
1.299072366877585719463141430187516699409725288457885838899818923753290643107951245283574655396808381
4
8/9:
1.445616095140239952072718914467638749495268402565399677644614166992051212768192453714749044140042691
4
1/10:
1.019851729415091843526358207816823199616111797408573908950154741278999619171876569877757774802125548
4
3/10:
1.067958034329354309181998720508432373228618500815506410559014146863362839020747504379185686844154080
4
7/10:
1.237040972195965067507214627505413280493090366952305815886395676403567796300574398662733422049912894
4
9/10:
1.468223828302126866617020254556201986284671438655060147322921731084641754120807497000027340805666758
4
1/11:
1.017949808347388069925036572359200727407752788316734278788094849366293932486562079862224350120127926
4
2/11:
1.037973201406283834010388116619437779198833483513001284322157753634176190170913031877999867010352099
4
3/11:
1.060575365795684179419177558655333970045651873777529084691070779601852013670678030311856368189927927
4
4/11:
1.086467887561676990139742092590604348519321188207866459128776367497925817044791201500433951265299498
4
5/11:
1.116698026349508783382702147897195456514259307666169063912927046836785925621828003799930846697807366
4
6/11:
1.152897723206490159822570818085382119128377594769358697681085613797160664950378163666265052887406530
4
7/11:
1.197815801592447954282731837806847275992433052785346468669525631954239479150874876033652459410353286
4
8/11:
1.256635262904252513562281658817613202870375685055264723198871769912858098988377594753529419465141380
4
9/11:
1.341065582053511880591970477711967466551467271249307970289504321908141258488247544857210299353208676
4
10/11:
1.488739273008270502192107813836297641770280602722820880684651859201619882121683149677142023450205425
4
1/12:
1.016380649862763958203394614866063321230008244031320683177072791718776652630354297598681855628065290
4
5/12:
1.103487869974153633944725282863562613939665018822798608741659105768526546189317103521986357702699759
4
7/12:
1.170335263092341515059322085723324457337091705314862698350708782562305665406496257251469613240716174
4
11/12:
1.507518925491628634840576929036616055503761041821412085912876421900979988317052502287004254688475145
4
1/13:
1.015063909112043910329752380504583903059464438272391531486117261888573627410852660958654795516086459
4
2/13:
1.031564186410085935173135737331878099385870517952401637712739394549158245926795950563146900405607148
4
3/13:
1.049783774091424383010647303440092262775864008209776123438693572375754346107864727470306903525945228
4
4/13:
1.070096700220435425892499826589912415135333313560023956808551484460632229700861961361356927209979356
4
5/13:
1.093011568273444772963269437028352254523975067159630530825877405210722395800371867640305278153241301
4
6/13:
1.119244432247447877841906313063072300948346695416013103256088960878978671145346388886432134617077626
4
7/13:
1.149848583299468819172943077113498301192583690193135915127451202389680159315719200432568580806346388
4
8/13:
1.186464446270064295666375458790106524030297355838317024063227559282358134284027170857759161353227208
4
9/13:
1.231853455545938714156193064467990317821071464742017700686016481548574516514409295087121217101383359
4
10/13:
1.291218894931713338827652749494816647691057485406891181871653422485270588239277250586598817041892313
4
11/13:
1.376307135657482924592192450772185058357818673163263599928126985312282547915025173374896033016835308
4
12/13:
1.524835046511076155602540022728405987120683510513104628242562281295236488917677223550085242271034972
4
1/14:
1.013943190010610564323069365018714891340672344472433639862476016680821169256855639810298422993269725
4
3/14:
1.045717783858293444612253646803884809179986266054983460344286256592559044560419334734025161822135360
4
5/14:
1.084489521945249411526110068428817616557387950235048484215492911201225480219367642258887379128767091
4
9/14:
1.201470758852427549563141481694525897783959575591285055972583578785336902192540696151300294420681609
4
11/14:
1.306653785149956055131564905728708845403685071650996315598578949673825932409392644518046942534707520
4
13/14:
1.540900315512067816254617757667889034636563591542932136588793977576850977256529472228443806751502672
4
1/15:
1.012977737938856382085263581337375655094269385866714039751601332228629289566047069207000081973140057
4
2/15:
1.027009046616660391852283354891676608010800216245294972502987745659210960317796857640207419643977891
4
4/15:
1.058975481776138096221097792770344937656599700639928735343871767592581307516719383679076759945654048
4
7/15:
1.121134462091537256427842569944089424741700371141939451023600948608562076059312044396227161920900272
4
8/15:
1.147643758955578211431188510126195522457603260223262453341236140381502004896107644308639396806817012
4
11/15:
1.261270130393394789092098072688134722301926607131945803534418923918290732524928064788967944661622676
4
13/15:
1.406683465745524759773908950699474888448585876066299675963867643112659006428329431299186908498125026
4
14/15:
1.555884066878087146814368912089640920790274333520447842513291995045858721979547517984163728435474162
4
1/16:
1.012137367060733176462299927611636013125359994659896107060886319856039353988470436256515216517970320
4
3/16:
1.039304134078761529582840906695057837640252595401395905537113719960589826517066781413326524358590519
4
5/16:
1.071446457779228805226322985801926241418530955036784835541726366202497175678267607042455273906685310
4
7/16:
1.110635235301763200702542122446537723935606163124924285263498596224636602324611204504538453324852304
4
9/16:
1.160544951462016932454733175000261816645311334561817275384423317659498002835009657358816494943907337
4
11/16:
1.228680393463767858290199626001947072811493551042769999759356473428706303808050322412930735322458443
4
13/16:
1.334601304190361232072079131804834788141976132565682292332583248381935931032347490886215561300376756
4
15/16:
1.569923340672879828887246087296189500744218272991636437585456864391674806767520192473278684305779026
4
1/17:
1.011399242051207171567588225304602405521765658548135657787843839221388751425994467261431989186269707
4
2/17:
1.023604210991689800175737782712501959665754884754635427032484369757822696293767683248747235486755173
4
3/17:
1.036729704281620421228264995310328681884428903557612872554900589926589462928117146476797806581947320
4
4/17:
1.050916480709315439958711202794782938590685714161821568982201463969865640187708075095938969286437578
4
5/17:
1.066339667634892566423708724414501190932214262704766334180011205327558513244428884215903502371268678
4
6/17:
1.083220858824981823094794750679191448021342453823907627136321282760653887855189009079926755141499052
4
7/17:
1.101846030539615173618068106309252783073517212489002281495365153388657923354673929871993641016382520
4
8/17:
1.122592985593691525166785971554570373241650673107693495643525980421373854618343881490298455051177916
4
9/17:
1.145975121345691355206833448720961362727094650393026873795713238551958600066369523540159441385633139
4
10/17:
1.172714734869645783873183528402748619992995778304189150321939987790599161295198519873118752122396603
4
11/17:
1.203873507323890269673664552453929497281081467914968906856250661796464675610739014718313278780081633
4
12/17:
1.241103605586037699767410878753893976186579326258099911860326919757846853763513054744522862902975119
4
13/17:
1.287183819116823295728644872444055517421992352190759789169327399931877478651850998590467272550544217
4
14/17:
1.347345160755997762761562331345099672789678164172440628175636137305921700188787731392723046427595997
4
15/17:
1.433384047764357371399787690532561006816142359671410223849535266511674240604434427245625375427141503
4
16/17:
1.583130618024089468931642255768420679808451487180278094626925361228592236216462137981702943875078027
4
1/18:
1.010745768573601894065122937832400829772710623219005141410981950076856279857681999321661722704186162
4
5/18:
1.061919672321873544246240104766115099464975520715709763116517309833839019256732467473741355161563466
4
7/18:
1.094374010156261421263662630282825589156544562110799302906731379667907781908566437723761448703815249
4
11/18:
1.184232896668945284215192335892627165000744054570136797692480399496607563647350931470910959053798973
4
13/18:
1.252855023593318167666725667997036295854141836587714021388142164252709089713721845663908527772755080
4
17/18:
1.595599368374051701171329726594287670791149599113004047155492780594869177649271042588268413366332470
4
1/19:
1.010163170966720441742114782366775378882924903350171498102004215948651585166238002130632223146931346
4
2/19:
1.020962439072687115177514408903460265369057580301291948980120708256618575411910361139376071040270823
4
3/19:
1.032477423171274855028635431770008247273722271826564523625304306074099361684538262274230265403643718
4
4/19:
1.044803417749542245259585853566466598523358692914925072322387085912193209496881956294348392144917550
4
5/19:
1.058055766447539604383508253418981575907993138430314688723732077998150765559012618748818437697717124
4
6/19:
1.072375869645616377782255371355230675508937713710853566708219845582477160589237678007867951145552204
4
7/19:
1.087939616514322802433134712649310027010031917811510950632176003161836943742394405777713399412532196
4
8/19:
1.104969516624095002959142700313506650515105310209320126126793004648677389137343991756739316523220297
4
9/19:
1.123752663109780950545044110588605897184741781317099924665094763069218195715632673476565621820144775
4
10/19:
1.144668244180308930738312641055225320433246837084003598885526072878530737443196865642964492625788604
4
11/19:
1.168231410950971530830429812793338510335613845336632273126910526641238954087751372447169008494429281
4
12/19:
1.195166736378336297295233781142770475045162934984172809921261430993267064203495267730734703910301182
4
13/19:
1.226538948312389494925108447709934359122821016396477930193586560503287908828918625712158211516569253
4
14/19:
1.264004459199210697517608309830616497005774796657017806011756148861612819911375669102444973663020970
4
15/19:
1.310348303093004108553238942610953031670646466518861204896084595748979198997117573845554415345475724
4
16/19:
1.370811414342962188754164994729194645687986289449342137542487423859420503230102124957740918869776538
4
17/19:
1.457207997882564508995080314847610928945438314811525344321559554139596275485689855109657848301292141
4
18/19:
1.607408112307335255176813272948171444767435180388043900213665796236914623416472896337718598659162230
4
1/20:
1.009640509362352553554163590498547414660266527206437593791136972747880999667432331277079176242990069
4
3/20:
1.030701004607039499617582087948955212070651477874390113901754265163955984938545731902520295104704227
4
7/20:
1.082338073263144500473831205653758182994484664857657347970658251212062998024714449287864234369772724
4
9/20:
1.115061589732827065338196878851480002604492268282666173172449649996900854691439409384538362193516897
4
11/20:
1.154906658383825711045063188161117689285321064289920495839402404092224265518907439979368147638866764
4
13/20:
1.205573555615801618729952409315006020104354192691429579253692537420185342695144626030347645398717037
4
17/20:
1.381669185212568089882110244428396789654846495655559904120696727254088215907445318453684477568887479
4
19/20:
1.618623452857967343673983386289841485251115958985900657894359182074004821192982460565693227071687752
6
1/2:
1.098430696839862068942935161608698822283938481692253723027033945469518521905843821564973446230505442
comment: A Gauss second-summation value, equal to $2/B((r+1)/(2r),(2r-1)/(2r))$.
6
1/3:
1.057076261613892128402122962965324995470501889069986725795375244685741777798779952700216143078965700
6
2/3:
1.157778216364893874950561674893505207806365917352306655090978043246823064194951989915747252390826364
6
1/4:
1.040343865043574145416878113346879640594465685268916549682386133655136588848837323908014059028010148
6
3/4:
1.200547022551705970623606133782633712417243715140519123069736520860927175815592908672157063181494441
6
1/5:
1.031227120754513755409097443523206049076896713820691129797578601885727886494785009328601926518885676
6
2/5:
1.072143911441301239285308516371365056707681862588226702509208055792540419288310570835512134058490225
6
3/5:
1.130948205380449462465758079793859332992924379485001810147213340696353596026888283755154647046083428
6
4/5:
1.234051513166650938678727539689680863939511554700611297922727915011535080521962974286976418925727248
6
1/6:
1.025479929962698069513640244088735839728532206675693532543652327980646441501018133834688540007781577
6
5/6:
1.261618848974220823232875404352187310613282634782723761417481556764687432263870744040189009223890045
6
1/7:
1.021522711978899270161484565462790266324217042630435591158386825263565261340839493875442725470508843
6
2/7:
1.047260452636646291936905526288772566333611423555714912707320878408599967268426922402365851159317897
6
3/7:
1.079152424402878754693764677906191997647926816101214401332204094757707475494039972314185499830646974
6
4/7:
1.120855392722023755059851942906939798671290619276856034395792254830210759062760167632958693523784511
6
5/7:
1.180632392947737505509653963634824163140583810153045236078522693791175570750097421002180002810962377
6
6/7:
1.285049626185067519149090613022787767841242600927099986053399274274054232969578961964855661848821311
6
1/8:
1.018630911495653293538424281488276717631384100968154539202986813100150927326279908118200895782928464
6
3/8:
1.066294964728936761384711034263120630667179508266381417209665290649859741745557242745148953209219164
6
5/8:
1.140419707762858011965213197113139193077077612250061736334808692654418410558010581934796829539301298
6
7/8:
1.305430277962542151058027584538875267811926497386817849305164939127018945600601626244266404389366698
6
1/9:
1.016424899916147157209316669882466843802873309665909634999036140962440390965478130838565284637450640
6
2/9:
1.035201978727051510335924015685416579908386169806777936956330730944242498718418183220770658796796304
6
4/9:
1.083207808460837506949863035558834733529058098651934955323580947078361815258019125807530880050500963
6
5/9:
1.115549034672973546023540352121884997137378434959801726832217629491907774083435314436783779577798586
6
7/9:
1.218198107849993176657400023408565088498766778634250505932396191626566673931547758165831938143491085
6
8/9:
1.323467526740777845709483066819782686939481648265356144701869051624955722200944233060569225543086875
6
1/10:
1.014686397293902488848773808132326995460963389822135686318875401550751735397643971017161747167955139
6
3/10:
1.050130505525393752543392435286971744933842913117885616261354068552253289500531337208548127076474095
6
7/10:
1.173382687647331618877433037707868835889088566992027544770349319783019806813872106817231525040982274
6
9/10:
1.339647244892691643646577789238076526892547089442876234566511576888432612311421152714764190141793242
6
1/11:
1.013280932066269384409544564443784330084997617455947494821181858444064525533770757194737037597871665
6
2/11:
1.028061371559877997235467812490300072031750441228370360755887709618727041100989702407199196956397666
6
3/11:
1.044703686625135244806701095695615524043133715665125128323460448674115255411632110119476910989154386
6
4/11:
1.063717679534189177003166853676704096970216661911781636153997305282625437224110951284829434586684943
6
5/11:
1.085852694109503816897090320356412402897529051463062715734787375429383093734800408496156128329462475
6
6/11:
1.112274884902382214336712525367454702412361695808694904263080889215295983248233775189017641356201706
6
7/11:
1.144945761804384534863307731283341063351054694044030220171593350941874301792650369229537129913809569
6
8/11:
1.187558294865867971535413640924808322601117613444200869507885567813962979236242844393906612869894116
6
9/11:
1.248442119965686532979394621641462256011781864276146780741931678886678906595056143104816242135589193
6
10/11:
1.354317975725572261605600824739944885470811372211539934333348402147269535242014756597437334730437889
6
1/12:
1.012121119795324153897807573801663641307652809534553826427844452770469902159869012438396264156873689
6
5/12:
1.076188188010642900354880843056866354684538866840620546257885412317167256633407561523339086022861796
6
7/12:
1.124972345770781377075516553607966658571863700199003283405019977138590132202245171453395465902092948
6
11/12:
1.367738326117550129350093100978188472084219150941623989490514206700619494670246097037363834143422482
6
1/13:
1.011147702553691862694693021558014559154490491290149126085507193567487746464627082515766385347153051
6
2/13:
1.023334374658149254548688827856581756461474427326541225514983127188458413704699316648116270739355181
6
3/13:
1.036763030181206511016844527818669882829844671857922468322467126566513246815626314549239792625168039
6
4/13:
1.051701768750087200685429003950656607320685730122428430809513767095178648450583056503607037247473832
6
5/13:
1.068514773331228135444705370243391939445667898134926564075424566306613610968893547754086606115864494
6
6/13:
1.087714217036116484688445110807252701985388909392262079944356346822890586944777065359045608603859188
6
7/13:
1.110052640964092956001916641473138728432129465741305720370197674110271022373261554099407297717057008
6
8/13:
1.136700692010361215596695232365626374065723932140552391472478042844049319311184405939467602924313235
6
9/13:
1.169626549972629532283177251533148163662249194575314750857445641587345802852172452327591140710112472
6
10/13:
1.212533898398616944333961540228208563643714338815146726378421068273264328230077708003401910715134613
6
11/13:
1.273771834048215942425478366739259707788219412148095640398202422940641618545318993378700453787114454
6
12/13:
1.380105511453614053314497268219894529620263277459598007612985700308707121718117411239914658715863237
6
1/14:
1.010319070885934624434439552540080807571397428374916818619278268569683813962583909348987246908531153
6
3/14:
1.033768677302322984544715739490180713547747393528869766676277001879443431314115650892434051669629775
6
5/14:
1.062266725118523737699974581730114990897962161417790867591020536694941037504428902068676356410316751
6
9/14:
1.147598993250444008603776381429957778606049706091985294763596824419770050403656206463539106619930418
6
11/14:
1.223663619654314575015761602111738900622791238034862748312575329594136212644473439167603710917237755
6
13/14:
1.391573394045489825786364213606628761100344362437273769802030498482527055534665599398942259849822130
6
1/15:
1.009605146659073866664932822431618252797819297973130104418322065981894500025458932970452613145019663
6
2/15:
1.019972512979176487339690497979029217934158708372584696725058892144768872510345229603238462335500274
6
4/15:
1.043527068146271087652514011463929507951811532073537755230302404416227367520100140018476456791953223
6
7/15:
1.089095611373551753435827059470324699843588466817184270042707209753173496317901457662627334649931872
6
8/15:
1.108445372203617562793553366265961346534596970061493373109708106776790531836017107653455877629034936
6
11/15:
1.190908699135308677282035139543141019548803590267457051817698267470191342226225244999885105611778275
6
13/15:
1.295570522690796564480299205190722046620505780257701036154485158100074624466470571682388039926908571
6
14/15:
1.402264341872106165712890646739097836281878847044122003941951024171928753773175776048263103363617459
6
1/16:
1.008983646056296893874646324104703632063855072826838286579924431755894153924039911316315740880452376
6
3/16:
1.029042558268967902210634220724062282098462628120319308655422582877080848890650673684466013200402936
6
5/16:
1.052693239175026916037849954621221549407762605478303192366315074577654045494861510209152314661179876
6
7/16:
1.081418721093007846249223515513129422271277655119663820149055467425504122775671861385753403919619810
6
9/16:
1.117845676110335647862815814685049740178090001859619227106398809115618081458948298215328986166848902
6
11/16:
1.167328346262450845135779688157952853841284653718080219381722403418542736503094733983467862925810994
6
13/16:
1.243791001963909424624897067178857385010242753326135283874963130479269920998845212382040092034370873
6
15/16:
1.412277287527642152662689677436275177905495197086003851122799215820176024981843900887880368086243980
6
1/17:
1.008437707778972006856380702708365390270763835945558531570917871581151519208505198023884699064556710
6
2/17:
1.017458409779463853659490725985483188727157665315033396992806041365955083556004913783650935401683465
6
3/17:
1.027144505133837577192574313239537879385252068105057352360135470667100007659203027575852416759744466
6
4/17:
1.037596951416296699498050364196781594820435300193483174920554796828830977838125176664088505021607129
6
5/17:
1.048941259669783293358204038025573429934101386480597573815260081173604675939017336532427642209804148
6
6/17:
1.061336116727804633228042505444906087981046484392830745860284526886768695983883753512060911249165911
6
7/17:
1.074986147671004018515884299579863739598055647851442032481433441680688575498226524219711759652312635
6
8/17:
1.090161455132572525091197411719743162959620380014954032686327552152624030325277141023358843320854494
6
9/17:
1.107228763813856173459842713412370372866050754147554825868240144824608347212552943601306508712896375
6
10/17:
1.126703554108313374947162755704263899048419634752544911926960130923783688260000739750289284789558789
6
11/17:
1.149342807714618362365980272619568667129926639644233390564490782632889269489980829143979905993604590
6
12/17:
1.176323380260667317526225279105929970533953728782115688826048190334928104784464921757865838182939616
6
13/17:
1.209622618655322876388464748791704479044716962797929811107844381598710024074563788230777547921120022
6
14/17:
1.252958836437306200628578380503590051947196305429924028974152226584443333719980669403369434050112693
6
15/17:
1.314707501075951743516029395516835418095088241999662729022559752013846576051328079705415815079789800
6
16/17:
1.421693361693456211075605744470633143642683523196311133246661424823759529309085806802476666617226378
6
1/18:
1.007954337850693614207843395061506267276748203138875027179452611499382664703312121107532674485190704
6
5/18:
1.045692181819062361824316665839975880224554276381188425430063329470725612127053791454354631379908834
6
7/18:
1.069513158919943451521272965405070039522480910614174809562912615109516022313932626592483476233359632
6
11/18:
1.135078932739293769446603158575339965992617301446784820512670682185602848160452304546771480428706890
6
13/18:
1.184824916410955393396465220235420385010855036533726676402920818476781175783105742696936726134562588
6
17/18:
1.430579928107982260608217337136958658569213832740413339503060014062614341829755000656702360273895023
6
1/19:
1.007523360742062216002023863813110535937309039817270141155721497870027203890565821369723962764061003
6
2/19:
1.015507027552438039960994847113426387123279831715781225649318904186679628652296403162472474366800792
6
3/19:
1.024008156253419755064581901164184928695991479904481342093475388049325416398449922231014864050127302
6
4/19:
1.033095109919926324684595375445722489161612498079168353572305877319364251532351617832612691489320735
6
5/19:
1.042850578001171643684263208459918995595856485793526350715019117203673333208995219660618141681860968
6
6/19:
1.053375858900291839380630226909900212399630344882456015484707115614452468228799686646541338302826146
6
7/19:
1.064796868101990141275554182181476524461657274023655227059779985267345251692590870197404396631221435
6
8/19:
1.077272778323969478894367265280502638781265368343893639885480400066318656650866698112032030732128863
6
9/19:
1.091008806814391660863204699684599384581814620790916187786206915533757954377173967196999926267724981
6
10/19:
1.106275790230422118770255036768391527798817095580892576225127963627376821923086013746449294509666988
6
11/19:
1.123441381787531973804418690634945144525154082330302797338991987106244848565558588760122071498684412
6
12/19:
1.143022265901048223337281449155958690187715706590249216676120205727000348872165807657850587175234179
6
13/19:
1.165777034762380773122793802898737063771420884647386871307019940014857599628115480475722925712068890
6
14/19:
1.192884786271960105148831144055601379085550872915327973392221703495348025104312123915560470596363366
6
15/19:
1.226326162298713915021298360914406985994434595872425963145977404643321288227416539753865625225325330
6
16/19:
1.269824706121067066901204596305931281728244906867673344319552113833143482501682979441606406745411945
6
17/19:
1.331765250607161445704679580437020298519246914977231304268550400388111855876065061627705236146021979
6
18/19:
1.438993535525749880880417382214506082740329792411949433088568852908326828910052379798504230205343183
6
1/20:
1.007136694362320855297515066969942871097850720730872470760483176446562716926106761826642502233849155
6
3/20:
1.022697455757682105127254417703997860089173176208858124283698786382509909176110791319109105806278018
6
7/20:
1.060688490091388067704201217291994405405828519017305873235576430858182552955313616856025058470060144
6
9/20:
1.084656154971247807414290135775368167630791189418175886523758962561412160244925754539302277722881481
6
11/20:
1.113738694252525242041295670796081209991090141681116551352411912781649906834096786714394649792298522
6
13/20:
1.150576439011235964792882965551513607590184353282936060439318541754878196867199879049683818289979105
6
17/20:
1.277621963605922934342149087711348852167024528004346008823060001311214976458555818641812891390662048
6
19/20:
1.446982117982091449688403200842536797386601725787857737480753842321516489423287027526302127497989144
Definition
This table gives the real values $F_r(x)={}_2F_1(1/r,1-1/r;1;x)$ of Gauss's hypergeometric function [3] for signatures $r\in\{2,3,4,6\}$ and rational arguments $0<x<1$. The argument $x$ is Gauss's hypergeometric variable.
Parameters
$r$
—   signature ($r\in\{2,3,4,6\}$)
$x$
—   hypergeometric argument ($0<x<1$)
Formulas
(1)
With $(a)_n$ denoting the Pochhammer symbol, $F_r(x)=\sum_{n=0}^{\infty} ((1/r)_n(1-1/r)_n/(n!)^2)x^n$ for $|x|<1$ [3].
(2)
For $y=1-x$ and $0<x<1$, $F_r(x)=\frac{\sin(\pi/r)}{\pi}\sum_{n=0}^{\infty} \frac{(1/r)_n(1-1/r)_n}{(n!)^2} \bigl(2\psi(n+1)-\psi(n+1/r)-\psi(n+1-1/r)-\log y\bigr)y^n$, where $\psi$ is the digamma function. This is the $c=a+b$ case of the continuation formula [5].
(3)
As $x\to1^-$, $F_r(x)/(-\log(1-x))\to\sin(\pi/r)/\pi$.
(4)
$F_2(x)=2K(x)/\pi$, where $K$ is the complete elliptic integral of the first kind with parameter $m=x$.
(5)
$F_2(1-b^2)=1/\operatorname{AGM}(1,b)$. The cubic analogue is $F_3(1-b^3)=1/\operatorname{AGM}_3(1,b)$, where $\operatorname{AGM}_3$ is the common limit of $a_{n+1}=(a_n+2b_n)/3$ and $b_{n+1}=\sqrt[3]{b_n(a_n^2+a_nb_n+b_n^2)/3}$ [2].
(6)
$F_4(x)=\sqrt{1+z}\,F_2(z)$ with $z=(1-\sqrt{1-x})/(1+\sqrt{1-x})$ [5].
(7)
For $r\in\{2,3,4,6\}$, $F_r(1/2)=2/B((r+1)/(2r),(2r-1)/(2r))= \sqrt{\pi}/(\Gamma((r+1)/(2r))\Gamma((2r-1)/(2r)))$ [4], with $B(a,b)$ as in [6].
Comments
(8)
Signature $2$ is the classical elliptic-integral case, related to $K(m)$ by (4). Signatures $3$, $4$ and $6$ are the alternative-base theories of Ramanujan developed by Berndt, Bhargava and Garvan [1]; in these four cases the two upper hypergeometric parameters are $a$ and $1-a$, with $a=1/r$.
(9)
$F_r(0)=1$, and $F_r(x)$ diverges logarithmically as $x\to1^-$ by (3).
(10)
At $x=1/2$, Gauss's second summation theorem gives the Gamma and Beta quotients in (7). The signature-$2$ entry agrees with [7]. The OEIS entry [8] writes the signature-$3$ value as $2^{-2/3}/L$, with $L=\Gamma(1/3)\Gamma(5/6)/\Gamma(1/6)$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.rational_field import QQ

r = 3
x = QQ(1) / QQ(2)
CBF = ComplexBallField(numberdb.bits(100, losing=128))
a = CBF(QQ(1) / QQ(r))
# For r in {3, 6} and x > 3/4 among the rows here, this arb call returns NaN
# because 1/r is not an exact binary ball; summing the logarithmic connection
# series in 1 - x is the method used for those rows.
CBF(x).hypergeometric([a, 1 - a], [CBF(1)]).real()
References
[1]
Bruce C. Berndt, S. Bhargava and Frank G. Garvan, Ramanujan's theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc. 347 (1995), no. 11, 4163-4244. (doi) (zbMATH) (MR)
[2]
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc. 323 (1991), no. 2, 691-701. (doi) (MR)
Links
Similar tables
Complete elliptic integral of the first kind $K(m)$ —   signature $2$ is related to $K(m)$ by (4)
Values of the arithmetic-geometric mean —   signature $2$ has the AGM expression and signature $3$ has the cubic analogue in (5)
Values of the Beta function $B(a,b)$ at pairs of rational numbers —   the entries with $x=1/2$ are reciprocal Beta values by (7)
Values of the Gamma function at rational numbers —   the entries with $x=1/2$ are Gamma quotients by (7)
Data properties
Entries are of type: real number
How well the digits are known: proven
How they were obtained:

Each value was computed as a Sage real ball with arb at numberdb.bits(digits, losing=128) bits. For $x\leq1/2$ the generator sums the Gauss series in (1) and adds the tail bound $x^N/(1-x)$, using $0<(a)_n(1-a)_n/(n!)^2\leq1$ for $0<a<1$.

more

For $x>1/2$ it sums the connection series in (2) and bounds the tail by a geometric series in $1-x$ times the largest digamma bracket, which occurs at $n=0$. The measured worst cases used 433 Gauss terms and 402 connection-series terms; the widest returned ball had radius less than $6\cdot10^{-130}$ for 100 requested digits. The 508 entries were checked against arb's hypergeometric function on the 448 rows where that call returned a finite value. On the remaining 60 rows, which have $r\in\{3,6\}$ and $x>3/4$, arb's direct hypergeometric call returns NaN because $1/r$ is not an exact binary ball, so the generator uses the connection series in (2) there. The entries were also checked against (4) for all 127 signature-$2$ rows, against (6) for all 127 signature-$4$ rows, against the cubic AGM in (5) for all 127 signature-$3$ rows, against the corresponding signature-$6$ quadratic transformation for all 64 rows with $x\leq1/2$, and against the Gamma and Beta values in (7) for all four signatures at $x=1/2$.

Table is complete: no (it holds every entry with $r\in\{2,3,4,6\}$ and $x=a/b$ in lowest terms with $1\leq a<b$ and $b\leq20$)


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