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132458 bytes, as of the version from 2026-10-26 04:16 (current). Recorded here, not run.
{
"Title": "Values of the Gamma function at rational numbers",
"Definition": "Listed are values of the Gamma function $\\Gamma(s)$ at rational arguments $s$.\nFor $\\Re(s) > 0$, $\\Gamma(s)$ is defined as the Mellin transform of $e^{-t}$, that is, $\\Gamma(s) = \\int_0^\\infty t^s e^{-t}\\, (dt/t)$. $\\Gamma$ extends to a meromorphic function on $\\mathbb{C}$ via analytic continuation.",
"Parameters": {
"s": {
"type": "Q",
"constraints": "$s \\not \\in \\mathbb{Z}_{\\leq 0}$"
}
},
"Comments": {
"comment-analytic-continuation": "The analytic continuation of $\\Gamma$ can be deduced from CITE{formula-functional-equation}."
},
"Formulas": {
"formula-functional-equation": "$\\Gamma(s+1) = s\\Gamma(s)$ for all $s\\in\\mathbb{C}$.",
"formula-factorial": "$\\Gamma(n) = (n-1)!$ for positive integers $n$.",
"formula-Euler-reflection": "$\\Gamma(s)\\Gamma(1-s) = \\pi / \\sin(\\pi s)$ (Euler's reflection formula)."
},
"Programs": {
"program-sage": {
"language": "Sage",
"code": "R = RealIntervalField(680) # 100 digits, with room\ns = R(7) / R(11)\ns.gamma()"
}
},
"References": {},
"Links": {
"Wiki": {
"title": "Wikipedia: Gamma function",
"url": "https://en.wikipedia.org/wiki/Gamma_function"
}
},
"Similar tables": [
{
"table": "HREF{Values_of_the_digamma_function_at_rational_numbers}[Values of the digamma function $\\psi(x)$ at rational numbers]",
"relation": "holds $\\psi=\\Gamma'/\\Gamma$, the logarithmic derivative of $\\Gamma$, at rational arguments"
},
{
"table": "HREF{Values_of_the_polygamma_functions_at_rational_numbers}[Values of the polygamma functions $\\psi^{(n)}(x)$ at rational numbers]",
"relation": "holds higher derivatives of $\\psi=\\Gamma'/\\Gamma$ at rational arguments"
},
{
"table": "HREF{Values_of_the_Beta_function_at_pairs_of_rational_numbers}[Values of the Beta function $B(a,b)$ at pairs of rational numbers]",
"relation": "uses $B(a,b)=\\Gamma(a)\\Gamma(b)/\\Gamma(a+b)$ to build beta values from three Gamma values"
},
{
"table": "HREF{Values_of_the_upper_incomplete_gamma_function}[Values of the upper incomplete gamma function $\\Gamma(a,x)$]",
"relation": "holds $\\Gamma(a,x)$, one term of $\\gamma(a,x)+\\Gamma(a,x)=\\Gamma(a)$"
},
{
"table": "HREF{Values_of_the_lower_incomplete_gamma_function}[Values of the lower incomplete gamma function $\\gamma(a,x)$]",
"relation": "holds $\\gamma(a,x)$, one term of $\\gamma(a,x)+\\Gamma(a,x)=\\Gamma(a)$"
},
{
"table": "HREF{Factorial}[Factorial of natural numbers]",
"relation": "holds $n!=\\Gamma(n+1)$ as exact integers for non-negative integers $n$"
},
{
"table": "HREF{P-adic_Gamma_function_at_integers}[Values of the $p$-adic Gamma function at integers]",
"relation": "holds values $\\Gamma_p(k)$ of the $p$-adic analogue at integers; this is not a restriction of the complex Gamma function"
}
],
"Keywords": [
"Euler gamma function",
"Euler integral of the second kind",
"factorial function"
],
"Tags": [
"special values",
"special functions"
],
"Data properties": {
"type": "R",
"complete": "no",
"rigour": "proven",
"rigour details": "The attached generator computed each row in Sage's `RealIntervalField(680)`, forming each rational argument as `RIFprec(a)/RIFprec(b)` so the interval contains the exact rational, then applying `gamma` and writing at most 100 digits. The stored inexact rows were checked against CITE{formula-Euler-reflection} where both $s$ and $1-s$ are present: 387 pairs, no failures; shifting $\\Gamma(1/3)$ by ten units in the last written place makes that check fail.",
"complete-note": "it holds $\\Gamma(a/b)$ for every fraction $a/b$ in lowest terms with denominator at most 30 and numerator at most 30 in absolute value, excluding the poles at the non-positive integers; for denominator 2 the arguments run from $-29/2$ to $29/2$, while for denominator 30 they run from $-29/30$ to $29/30$"
},
"Display properties": {
"number-header": "$\\Gamma(s)$"
},
"Numbers": {
"1": {
"proof": "CITE{formula-factorial}",
"number": "1",
"equals": "HREF{Factorial#0}[0!]"
},
"2": {
"proof": "CITE{formula-factorial}",
"number": "1",
"equals": "HREF{Factorial#1}[1!]"
},
"3": {
"proof": "CITE{formula-factorial}",
"number": "2",
"equals": "HREF{Factorial#2}[2!]"
},
"4": {
"proof": "CITE{formula-factorial}",
"number": "6",
"equals": "HREF{Factorial#3}[3!]"
},
"5": {
"proof": "CITE{formula-factorial}",
"number": "24",
"equals": "HREF{Factorial#4}[4!]"
},
"6": {
"proof": "CITE{formula-factorial}",
"number": "120",
"equals": "HREF{Factorial#5}[5!]"
},
"7": {
"proof": "CITE{formula-factorial}",
"number": "720",
"equals": "HREF{Factorial#6}[6!]"
},
"8": {
"proof": "CITE{formula-factorial}",
"number": "5040",
"equals": "HREF{Factorial#7}[7!]"
},
"9": {
"proof": "CITE{formula-factorial}",
"number": "40320",
"equals": "HREF{Factorial#8}[8!]"
},
"10": {
"proof": "CITE{formula-factorial}",
"number": "362880",
"equals": "HREF{Factorial#9}[9!]"
},
"11": {
"proof": "CITE{formula-factorial}",
"number": "3628800",
"equals": "HREF{Factorial#10}[10!]"
},
"12": {
"proof": "CITE{formula-factorial}",
"number": "39916800",
"equals": "HREF{Factorial#11}[11!]"
},
"13": {
"proof": "CITE{formula-factorial}",
"number": "479001600",
"equals": "HREF{Factorial#12}[12!]"
},
"14": {
"proof": "CITE{formula-factorial}",
"number": "6227020800",
"equals": "HREF{Factorial#13}[13!]"
},
"15": {
"proof": "CITE{formula-factorial}",
"number": "87178291200",
"equals": "HREF{Factorial#14}[14!]"
},
"16": {
"proof": "CITE{formula-factorial}",
"number": "1307674368000",
"equals": "HREF{Factorial#15}[15!]"
},
"17": {
"proof": "CITE{formula-factorial}",
"number": "20922789888000",
"equals": "HREF{Factorial#16}[16!]"
},
"18": {
"proof": "CITE{formula-factorial}",
"number": "355687428096000",
"equals": "HREF{Factorial#17}[17!]"
},
"19": {
"proof": "CITE{formula-factorial}",
"number": "6402373705728000",
"equals": "HREF{Factorial#18}[18!]"
},
"20": {
"proof": "CITE{formula-factorial}",
"number": "121645100408832000",
"equals": "HREF{Factorial#19}[19!]"
},
"21": {
"proof": "CITE{formula-factorial}",
"number": "2432902008176640000",
"equals": "HREF{Factorial#20}[20!]"
},
"22": {
"proof": "CITE{formula-factorial}",
"number": "51090942171709440000",
"equals": "HREF{Factorial#21}[21!]"
},
"23": {
"proof": "CITE{formula-factorial}",
"number": "1124000727777607680000",
"equals": "HREF{Factorial#22}[22!]"
},
"24": {
"proof": "CITE{formula-factorial}",
"number": "25852016738884976640000",
"equals": "HREF{Factorial#23}[23!]"
},
"25": {
"proof": "CITE{formula-factorial}",
"number": "620448401733239439360000",
"equals": "HREF{Factorial#24}[24!]"
},
"26": {
"proof": "CITE{formula-factorial}",
"number": "15511210043330985984000000",
"equals": "HREF{Factorial#25}[25!]"
},
"27": {
"proof": "CITE{formula-factorial}",
"number": "403291461126605635584000000",
"equals": "HREF{Factorial#26}[26!]"
},
"28": {
"proof": "CITE{formula-factorial}",
"number": "10888869450418352160768000000",
"equals": "HREF{Factorial#27}[27!]"
},
"29": {
"proof": "CITE{formula-factorial}",
"number": "304888344611713860501504000000",
"equals": "HREF{Factorial#28}[28!]"
},
"30": {
"proof": "CITE{formula-factorial}",
"number": "8841761993739701954543616000000",
"equals": "HREF{Factorial#29}[29!]"
},
"-29/2": "-9.382408602090834818133636560604108398005329539754878671097049358974390640171788499733360062393406905e-12",
"-27/2": "1.360449247303171048629377301287595717710772783264457407309072157051286642824909332461337209047044001e-10",
"-25/2": "-1.836606483859280915649659356738254218909543257407017499867247412019236967813627598822805232213509401e-9",
"-23/2": "2.295758104824101144562074195922817773636929071758771874834059265024046209767034498528506540266886752e-8",
"-21/2": "-2.640121820547716316246385325311240439682468432522587656059168154777653141232089673307782521306919765e-7",
"-19/2": "2.772127911575102132058704591576802461666591854148717038862126562516535798293694156973171647372265753e-6",
"-17/2": "-0.000026335215159963470254557693619979623385832622614412811869190202343907090083790094491245130650036",
"-15/2": "0.000223849328859689497163740395769826798779577292222508900888116719923210265712215803175583610525310",
"-13/2": "-0.001678869966447671228728052968273700990846829691668816756660875399424076992841618523816877078939828",
"-11/2": "0.010912654781909862986732344293779056440504392995847308918295690096256500453470520404809701013108884",
"-9/2": "-0.060019601300504246427027893615784810422774161477160199050626295529410752494087862226453355572098866",
"-7/2": "0.270088205852269108921625521271031646902483726647220895727818329882348386223395380019040100074444901",
"-5/2": "-0.945308720482941881225689324448610764158693043265273135047364154588219351781883830066640350260557154",
"-3/2": "2.363271801207354703064223311121526910396732608163182837618410386470548379454709575166600875651392887",
"-1/2": "-3.544907701811032054596334966682290365595098912244774256427615579705822569182064362749901313477089330",
"1/2": "1.772453850905516027298167483341145182797549456122387128213807789852911284591032181374950656738544665",
"3/2": "0.886226925452758013649083741670572591398774728061193564106903894926455642295516090687475328369272332",
"5/2": "1.329340388179137020473625612505858887098162092091790346160355842389683463443274136031212992553908499",
"7/2": "3.323350970447842551184064031264647217745405230229475865400889605974208658608185340078032481384771247",
"9/2": "11.63172839656744892914422410942626526210891830580316552890311362090973030512864869027311368484669936",
"11/2": "52.34277778455352018114900849241819367949013237611424488006401129409378637307891910622901158181014715",
"13/2": "287.8852778150443609963195467083000652371957280686283468403520621175158250519340550842595636999558093",
"15/2": "1871.254305797788346476077053603950424041772232446084254462288403763852862837571358047687164049712760",
"17/2": "14034.40729348341259857057790202962818031329174334563190846716302822889647128178518535765373037284570",
"19/2": "119292.4619946090070878499121672518395326629798184378712219708857399456200058951740755400567081691884",
"21/2": "1.133278388948785567334574165588892475560298308275159776608723414529483390056004153717630538727607290e6",
"23/2": "1.189942308396224845701302873868337099338313223688917765439159585255957559558804361403512065663987655e7",
"25/2": "1.368433654655658572556498304948587664239060207242255430255033523044351193492625015614038875513585803e8",
"27/2": "1.710542068319573215695622881185734580298825259052819287818791903805438991865781269517548594391982254e9",
"29/2": "2.309231792231423841189090889600741683403414099721306038555369070137342639018804713848690602429176043e10",
"-29/3": "2.177642028136199693294058943850897044354729447384245978110513955737998352169193812185856004705470096e-6",
"-28/3": "4.692165811467052579716002557973974129790014715863765645566172121510621681185795838525685463647906219e-6",
"-26/3": "-0.000021050539605316597035175903123892004762095717991381044455068301572133984070968873517796608045486",
"-25/3": "-0.000043793547573692490744016023874423758544706804014728479358617606467432469024400761159573064327380",
"-23/3": "0.000182438009912743840971524493740397374604829555925302385277258613625161195281730237154237269727547",
"-22/3": "0.000364946229780770756200133532286864654539223366789403994655146720561937241870006342996442202728170",
"-20/3": "-0.001398691409331036114115021118676379871970359928760651620458982704459569163826598484849152401244528",
"-19/3": "-0.002676272351725652212134312570103674133287638023122295960804409284120873107046713181973909486673250",
"-17/3": "0.009324609395540240760766807457842532479802399525071010803059884696397127758843989898994349341630188",
"-16/3": "0.016949724894262464010183979610656602844155040813107874418427925466098863011295850152501426748930584",
"-14/3": "-0.052839453241394697644345242261107684052213597308735727884006013279583723966782609427634646269237733",
"-13/3": "-0.090398532769399808054314557923501881835493551003241996898282269152527269393577867480007609327629784",
"-11/3": "0.246584115126508589006944463885169192243663454107433396792028061971390711844985510662295015923109422",
"-10/3": "0.391726975334065834902029751001841487953805387680715319892556499660951500705504092413366307086395734",
"-8/3": "-0.904141755463864826358796367578953704893432665060589121570769560561765943431613539095081725051401216",
"-7/3": "-1.305756584446886116340099170006138293179351292269051066308521665536505002351680308044554356954652449",
"-5/3": "2.411044681236972870290123646877209879715820440161570990855385494831375849150969437586884600137069910",
"-4/3": "3.046765363709400938126898063347656017418486348627785821386550552918511672153920718770626832894189049",
"-2/3": "-4.018407802061621450483539411462016466193034066935951651425642491385626415251615729311474333561783184",
"-1/3": "-4.062353818279201250835864084463541356557981798170381095182067403891348896205227625027502443858918733",
"1/3": "2.678938534707747633655692940974677644128689377957301100950428327590417610167743819540982889041188789",
"2/3": "1.354117939426400416945288028154513785519327266056793698394022467963782965401742541675834147952972911",
"4/3": "0.892979511569249211218564313658225881376229792652433700316809442530139203389247939846994296347062929",
"5/3": "0.902745292950933611296858685436342523679551510704529132262681645309188643601161694450556098635315274",
"7/3": "1.190639348758998948291419084877634508501639723536578267089079256706852271185663919795992395129417239",
"8/3": "1.504575488251556018828097809060570872799252517840881887104469408848647739335269490750926831058858790",
"10/3": "2.778158480437664212679977864714480519837159354918682623207851598982655299433215812857315588635306892",
"11/3": "4.012201302004149383541594157494855660798006714242351698945251756929727304894051975335804882823623440",
"13/3": "9.260528268125547375599926215714935066123864516395608744026171996608850998110719376191051962117689642",
"14/3": "14.71140477401521440631917857748113742292602461888862289613258977540900011794485724289795123701995261",
"16/3": "40.12895582854403862759968026809805195320341290438097122411341198530502099181311729682789183584332178",
"17/3": "68.65322227873766722948950002824530797365478155481357351528541895190866721707600046685710577275977886",
"19/3": "214.0210977522348726805316280965229437504182021566985131952715305882934452896699589164154231244977161",
"20/3": "389.0349262461801143004405001600567451840437621439435832532840407274824475634306693121902660456387469",
"22/3": "1355.466952430820860310033644611311977085981946992423916903386360392525153501243073137297679788485535",
"23/3": "2593.566174974534095336270001067044967893625080959623888355226938183216317089537795414601773637591646",
"25/3": "9940.090984492686308940246727149621165297200944611108723958166642878517792342449203006849651782227262",
"26/3": "19884.00734147142806424473667484734475385112562069044981072340652607132509768645643151194693122153595",
"28/3": "82834.09153743905257450205605958017637747667453842590603298472202398764826952041002505708043151856052",
"29/3": "172328.0636260857098901210511820103212000430887126505650262695232259514841799492890731035400705866449",
"-29/4": "0.000530397706352147861852210714986219897896379614077825515392637005928414706606788285586458387123403",
"-27/4": "-0.001452956373178291597893194548703675895371396512466619939259862561907955290219853491842014455943595",
"-25/4": "-0.003845383371053071998428527683650094259748752202064234986596618292981006622899215070501823306644673",
"-23/4": "0.009807455518953468285779063203749812293756926459149684590004072292878698208984011069933597577619266",
"-21/4": "0.024033646069081699990178298022813089123429701262901468666228864331131291393120094190636395666529206",
"-19/4": "-0.056392869233982442643229613421561420689102327140110686392523415684052514701658063652118186071310784",
"-17/4": "-0.126176641862678924948436064619768717898005931630232710497701537738439279813880494500841077249278333",
"-15/4": "0.267866128861416602555340663752416748273236053915525760364486224499249444832875802347561383838726228",
"-13/4": "0.536250727916385431030853274634017051066525209428489019615231535388366939208992101628574578309432918",
"-11/4": "-1.004497983230312259582527489071562806024635202183221601366823341872185418123284258803355189395223355",
"-9/4": "-1.742814865728252650850273142560555415966206930642589313749502490012192552429224330292867379505656983",
"-7/4": "2.762369453883358713851950594946797716567746806003859403758764190148509899839031711709226770836864228",
"-5/4": "3.921333447888568464413114570761249685923965593945825955936380602527433242965754743158951603887728213",
"-3/4": "-4.834146544295877749240913541156896003993556910506753956577837332759892324718305495491146848964512400",
"-1/4": "-4.901666809860710580516393213451562107404956992432282444920475753159291553707193428948689504859660266",
"1/4": "3.625609908221908311930685155867672002995167682880065467433377999569919243538729121618360136723384300",
"3/4": "1.225416702465177645129098303362890526851239248108070611230118938289822888426798357237172376214915066",
"5/4": "0.906402477055477077982671288966918000748791920720016366858344499892479810884682280404590034180846075",
"7/4": "0.919062526848883233846823727522167895138429436081052958422589203717367166320098767927879282161186299",
"9/4": "1.133003096319346347478339111208647500935989900900020458572930624865599763605852850505737542726057593",
"11/4": "1.608359421985545659231941523163793816492251513141842677239531106505392541060172843873788743782076024",
"13/4": "2.549256966718529281826263000219456877105977277025046031789093905947599468113168913637909471133629586",
"15/4": "4.422988410460250562887839188700432995353691661140067362408710542889829487915475320652919045400709068",
"17/4": "8.285085141835220165935354750713234850594426150331399603314555194329698271367798969323205781184296155",
"19/4": "16.58620653922593961082939695762662373257634372927525260903266453583686057968303245244844642025265900",
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