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Values of the upper incomplete gamma function $\Gamma(a,x)$
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Numbers
$a$
$x$ 
$\Gamma(a,x)$
1/2
1/2:
0.5624182315944071242794949573020430690267675647965080152711827351225288269777202721155730664137029081
1/2
1/4:
0.8498918380799311297867616098602389766300312781938048348528626579829435160004444679496261593386597163
1/2
1:
0.2788055852806619764992326110774391720885500824971744701584987919357006294674865183792537970927408260
1/2
2:
0.08064711796031769078862607302130517570135929216816690059672139112255856816240862110577321725817665903
1/2
9/2:
0.004785263928985074244974471845154161395022898634542486927911346086170450862093086078175392049618291943
1/2
8:
0.0001122716291001465580445381746990559362941551668835616835235680696367564132132596432171648208299220524
1/2
25/2:
0.000001016153364890793152459050077989108978142268608759972263610976110046875938969400308179671893939608750
1/2
18:
3.497362141405744779078158741209820620655332508232029602140253966272941621214564611847687607647936246e-9
1
1:
0.3678794411714423215955237701614608674458111310317678345078368016974614957448998033571472743459196437
1
1/2:
0.6065306597126334236037995349911804534419181354871869556828921587350565194137484239986476115079894560
1
2:
0.1353352832366126918939994949724844034076315459095758814681588726540733741014876899370981224906570488
1
0:
1
3/2
3/2:
0.3470687758966215683176335659423909812092738484707891795933076936000120771780433907417756010372449596
3/2
3/4:
0.6046463371814992427597554810240427154615892586154760069760950722937197940909302944030726615562961531
3/2
3:
0.09891198663477736960412933549471480504001012012535334105840203296243307783046054886477725197651195295
3/2
1:
0.5072822338117733098451400757001804534900861722803550695870861976653118104786430625467741728922900567
2
2:
0.4060058497098380756819984849174532102228946377287276444044766179622201223044630698112943674719711463
2
1:
0.7357588823428846431910475403229217348916222620635356690156736033949229914897996067142945486918392875
2
4:
0.09157819444367090146859010636620621105956033776737797384799963719625215799057061343684749064451025865
5/2
5/2:
0.5528463292166205905792788376365536675643919962115273794539885042516431082102881299747776650263573805
5/2
5/4:
1.032226259266264416260747733540102054814380561937348292230356740574832314433238510201774647488287089
5/2
5:
0.1000132513171574223384170211790580129187651315579454574298738109356723077243946927587572564215041764
5/2
1:
1.128802791889102286363233883711731547680940389452300438888466098195429211462864397177308533684354729
3
3:
0.8463801622536870306488210660510502027388930672031946646496713693030313514133923526189221520227668569
3
3/2:
1.617693661076116259766283413039090779730744312867825133392806061012357458077029698537337044911187533
3
6:
0.1239376088333179211522583715408333945753239792766972525439312003138097114146995091187360200791333909
3
1:
1.839397205857211607977618850807304337229055655158839172539184008487307478724499016785736371729598219
7/2
7/2:
1.425318290804476877962038789101062769479412509594861708922903545672153689080361298721525286804382750
7/2
7/4:
2.775747418175848181605701084683268862154908308070368947773921592745427392483400432216958682269417016
7/2
7:
0.1700936005341450666731762563105253645509954115250473883838295030119900236550136313414926026857110706
7/2
1:
3.189886420894198037503608479440789736648162104662518931729002047186034524402060796300418608556806466
4
4:
2.600820722200253601707959020800256394091513592593534457283189696373561286932205421606468734304091346
4
2:
5.142740762991282291971980808954407329489998744563883495790037160854788215856532217609728654644967853
4
8:
0.2542806719501039738266129573418926526376623013600020752927884169170235146971622530387426873117045193
4
1:
5.886071058743077145528380322583373879132978096508285352125388827159383931918396853714356389534714300
9/2
9/2:
5.086254600275426697913639521658010730996650663525051864372270593095170813752910875972462124348963572
9/2
9/4:
10.18403214286566234526019345802929635910911247967051220522868706375276166311515850153711303629382085
9/2
9:
0.4091290578259129167084715853082353502626692647486120256202132894658723311868270942796421924196065643
9/2
1:
11.53248191430113545285815344820422494571437849735058409555934396684858233115211259040861240429474227
5
5:
10.57183884156509787462195997591987764644499890792012904014937903754044682698277371959369884656291927
5
5/2:
21.38827245393962981636031524084932742973783628208471181487292583987325071465208371189714385450217665
5
10:
0.7020645138470657441463871966283546367191653262325606846222786705870550558435704834746466702985365058
5
1:
23.91216367614375090370904506049495638397772351706490924300939211033499722341848721821457283248477684
11/2
11/4:
47.34721093889289216806344675250821159244248501089256671624627887495395856392955773392181319974257050
11/2
11/2:
23.20163128272915032444991355633586638235565172251216718892553613239464367760718753577411311963362893
11/2
11:
1.275763281368427237650758597030442862733409282171446655627141048761216235628440650684808073225523375
11/2
1:
52.26404805552655185945721428708047312316051436910939626452488465251608198592940646019590309367225988
6
3:
109.9298469562435860983880537553364028027926995520384599733220225541819543482806067401517712744864247
6
6:
53.48155696375334933562253248730042642714380353748039841775719155941516666967113217491696738454764084
6
12:
2.440923530031404547251237934810431432344576683104286036110051466568861748723531161363155931623153503
6
1:
119.9286978218901968401407490726362427873344287163563140495547973533724476128373358944300114367698039
13/2
13/4:
266.5964042924957708737533986125867276565208593668262071140467399858051718676791336305231627451243536
13/2
13/2:
128.9183882702869789665782823936162561533749074769607216528771670117633715651571459917728453234243721
13/2
13:
4.894290647225386117072822124748968653564953166037648799202929917753674364602213725775603136466414863
13/2
1:
287.8201437465674775486102023491040630448286401611334472893947023905359124183566353344346142895433490
7
7/2:
672.9925701391533455575250272793591325093247178499587702586361627118921703011110300328540561272552870
7
7:
323.7919602110631696539615639881324842909165789265214338508334715779384085761929527397044758945633139
7
14:
10.24410120786831490451467136234700536464850799326394300724731732542361671257218061308561353231333209
7
1:
719.9400663725126233624400182059789175914523834291696521318366209219321471727689151699372158949647428
15/2
15/4:
1763.213908264702989661575370949300715059848075172056585436266999935766825150304571077130592224402733
15/2
15/2:
844.7164002173680648398224050770897277173027234132860085720041201998887718144369365053867776340639799
15/2
15:
22.30815060583217396950708294011737441430351281159461815898256125270775596445224797573639501045722735
15/2
1:
1871.198813793860046387561839039337870658831972178399175215573402340180892215063029477182140156377688
8
4:
4782.286576404049411410950226528381306499088486482691696721619454098044691993989890162925439448841315
8
8:
2282.922479814452206683844132119990371272391436406354463036559158960095589016542865376706405002102267
8
16:
50.39889600364814951645253614610661239540244964505858463413380777137799379522535410107914422901709840
8
1:
5039.948344048759805858675651212013884007612495135219332757364183255222491705127305992917658539099119
17/2
17/4:
13398.02611556486261656519607014407040013307010409888898043369516324190546471673442203096306351172661
17/2
17/2:
6376.759101295021456337303820746054585087015067521626463802044215041631715663748360323703460292800449
17/2
17:
117.8346090840893084178053642198781750440584425703207167706619826572853421777546878511803698206227762
17/2
1:
14034.35898289512179022830931656519549080868560246902558195130835435305415310871762088222319844717858
9
9/2:
38696.82516072423952355706046040199814503747029763913999664139325564202231313259644949331750409249237
9
9:
18371.91300627992311613474965070308734871070902940540801943084535447015551739359979803948312987599234
9
18:
284.4982888269366160877517511252798455244028043822653104845308033312273987827926604521949427819652366
9
1:
40319.95463183124988919100073346627253292834577221278642989342130284347739513676334774669841558713887
19/2
19/4:
115024.3650809717829614557158952938535818609469706505067052555023433435215924265845226514591193862838
19/2
19/2:
54497.10614994359981692373110011290471586533423892262295015265476672079428074253466789176570238727107
19/2
19:
707.9660486906040395472808222701861163781685086586131482849216847342130649689694223356310410549828620
19/2
1:
119292.4192340497066592622247145743231327412734321177492144206288488026577629198446773022543340753638
10
5:
351330.2345647243979168097359612578316362425279838996851595539364288114115143024836011894410888777141
10
10:
166173.5347875457293530441433296913330263963194250345282207707751818606906636001235643483332311493533
10
20:
1812.735218438657230866762945975425877103438697704246733335760565658455915213852709445928150881839728
10
1:
362879.9595659224204450406021249666142572225577610461096368752995623929940177266150295236428875585958
Definition
For real $x>0$, the upper incomplete gamma function is $\Gamma(a,x)=\int_x^\infty t^{a-1}e^{-t}\,\mathrm{d}t$ [1]. This table gives $\Gamma(a,x)$ at rational parameters $a$ and positive rational arguments $x$. At $x=0$ and $a>0$ the convergent endpoint value is included.
Parameters
$a$
—   parameter ($a\in\mathbb{Q}$)
$x$
—   argument ($x\geq0$, with $a>0$ if $x=0$)
Formulas
(1)
If $a\ne0,-1,-2,\ldots$, then $\gamma(a,x)+\Gamma(a,x)=\Gamma(a)$, where $\gamma(a,x)$ is the lower incomplete gamma function: $\gamma(a,x)=\int_0^x t^{a-1}e^{-t}\,\mathrm{d}t$ for $a>0$, and it is defined for nonintegral negative $a$ by analytic continuation in $a$ [1].
(2)
$\Gamma(a+1,x)=a\Gamma(a,x)+x^ae^{-x}$ [3].
(3)
$\Gamma(0,x)=E_1(x)$ [2].
(4)
$\Gamma(1/2,x)=\sqrt{\pi}\operatorname{erfc}(\sqrt{x})$ [2].
(5)
For nonnegative integers $n$, $\Gamma(n+1,x)=n!e^{-x}\sum_{k=0}^n x^k/k!$ [2].
(6)
$\Gamma(a,x)=x^ae^{-x}U(1,a+1,x)$, where $U$ is Tricomi's confluent hypergeometric function [4].
Comments
(7)
For positive real $x$ and real $a$, the defining integral is real and agrees with the principal branch of $\Gamma(a,z)$ in the argument $z$.
(8)
The regularized upper incomplete gamma function is $Q(a,x)=\Gamma(a,x)/\Gamma(a)$ when $\Gamma(a)$ is finite [1]. For $a=1/2,1,3/2,2,5/2$, $Q(a,x)$ is the survival probability of a chi-squared random variable with $2a$ degrees of freedom at $2x$. The regularized value is not stored here.
(9)
For a chi-square variable with $n$ degrees of freedom, $\Gamma(n/2,x)/\Gamma(n/2)$ is its upper-tail probability at threshold $2x$ [9]. The half-integer shape family is extended beyond the transition $x=a$ to the unit scale and half/double mean thresholds. These are explicit reference comparisons, not quantiles or conventional significance levels. The shape $a=1/2$ also includes integer standard-normal-deviation thresholds $x=k^2/2$ [10]. The values remain unregularized integrals. No free Cartesian grid of shape and threshold is used.
Programs
(P1)
Python
from balanced_values import BalancedValues
from numberdb._write import to_text
reference = BalancedValues()
for parameters in reference.enumerate():
    print(parameters, to_text(reference.value(parameters, reference.digits), reference.digits))
Links
Similar tables
Values of the Gamma function at rational numbers —   for $a\ne0,-1,-2,\ldots$, the lower and upper incomplete gamma functions sum to $\Gamma(a)$
Values of the exponential integral —   the row $a=0$ is the exponential integral $E_1(x)$; for $1/10\leq x\leq5$, both tables store the same $E_1(x)$ values
Values of the complementary error function $\operatorname{erfc}(x)$ —   the row $a=1/2$ is $\sqrt{\pi}\operatorname{erfc}(\sqrt{x})$
Values of Tricomi's confluent hypergeometric function $U(a,b,z)$ —   $\Gamma(a,x)=x^ae^{-x}U(1,a+1,x)$, where the corresponding parameters of $U$ are $(1,a+1,x)$
Data properties
Entries are of type: real number
How well the digits are known: proven
How they were obtained:

All 83 stored entries are recomputed from exact rational or enclosed square-root inputs using Arb at 100 decimal digits plus 160 guard bits. Existing numeric strings and row annotations are retained unchanged; their written intervals strictly contain the fresh enclosures. Exact anchors and rational polynomial integrals use exact arithmetic.

more

Every inexact result is finite, with at least 458 relative accuracy bits. Independent mpmath computations at 150 and 200 decimal digits check every value; the AGM check uses its elliptic-integral expression, and the incomplete-gamma check uses integer/half-integer closed forms. Known nonzero and exact controls, shifted-last-place rejection controls, enumeration checks and applicable identities pass. The attached balanced-evidence.json records the checks, and balanced_checks.py repeats them. The current generate.py and balanced-selection.json reproduce the full retained-plus-added selection; older auxiliary attachments are preserved as historical sources, not claimed as the default enumeration.

Table is complete: no (it includes $a=n/2$ for $1\leq n\leq20$ at $x=a/2,a,2a,1$, with duplicate pairs merged; $a=1/2$ also has $x=k^2/2$ for $1\leq k\leq6$; the exact endpoint $\Gamma(1,0)=1$ is included; the dimensional and threshold bounds delimit reference families rather than a quantile table)


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