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Values of the lower incomplete gamma function $\gamma(a,x)$
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Numbers
$a$
$x$ 
$\gamma(a,x)$
1/2
1/2:
1.210035619311108903018672526039102113770781891325879112942625054730382457613311909259377590324841757
1/2
1/4:
0.9225620128255848975114058734809062061675181779285822933609451318699677685905877134253244973998849491
1/2
1:
1.493648265624854050798934872263706010708999373625212658055308997917210655123545662995696859645803839
1/2
2:
1.691806732945198336509541410319840007096190163954220227617086398730352716428623560269177439480368006
1/2
9/2:
1.767668586976530953053193011495991021402526557487844641285896443766740833728939095296775264688926373
1/2
8:
1.772341579276415880740122945166446126861255300955503566530284221783274528177818921731733491917714743
1/2
25/2:
1.772452834752151136505015024291067193688571313853778368241544178876801237715093211974642477066650726
1/2
18:
1.772453847408153885892422704262986441587728835467054619981778187712657318318090560160386044890857058
1
1:
0.6321205588285576784044762298385391325541888689682321654921631983025385042551001966428527256540803563
1
1/2:
0.3934693402873665763962004650088195465580818645128130443171078412649434805862515760013523884920105440
1
2:
0.8646647167633873081060005050275155965923684540904241185318411273459266258985123100629018775093429512
1
0:
0
3/2
3/2:
0.5391581495561364453314501757281816101895008795904043845135962013264435651174726999456997273320273731
3/2
3/4:
0.2815805882712587708893282606465298759371854694457175571308088226327358482045857962844026668129761796
3/2
3:
0.7873149388179806440449544061758577863587646079358402230485018619640225644650555418226980763927603798
3/2
1:
0.3789446916409847038039436659703921379086885557808384945198176972611438318168730281407011554769822760
2
2:
0.5939941502901619243180015150825467897771053622712723555955233820377798776955369301887056325280288537
2
1:
0.2642411176571153568089524596770782651083777379364643309843263966050770085102003932857054513081607125
2
4:
0.9084218055563290985314098936337937889404396622326220261520003628037478420094293865631525093554897414
5/2
5/2:
0.7764940589625164298943467748693052195337700958802629667063673381380403552329860060564353275275511186
5/2
5/4:
0.2971141289128726042128778789657568322837815301544420539299991018148511490100356258294383450656214102
5/2
5:
1.229327136861979598135208591326800874179396960533844888730482031454011155718879443272455736132404323
5/2
1:
0.2005375962900347341103917287941273394172217026394899072718897441942542519804097388539044588695537702
3
3:
1.153619837746312969351178933948949797261106932796805335350328630696968648586607647381077847977233143
3
3/2:
0.3823063389238837402337165869609092202692556871321748666071939389876425419229703014626629550888124669
3
6:
1.876062391166682078847741628459166605424676020723302747456068799686190288585300490881263979920866609
3
1:
0.1606027941427883920223811491926956627709443448411608274608159915126925212755009832142636282704017813
7/2
7/2:
1.898032679643365673222025242163584448265992720634614156477986060302054969527824041356507194580388498
7/2
7/4:
0.5476035522719943695783629465813783555904969221591069176269680132287812661247849078610737991153542321
7/2
7:
3.153257369913697484510887774954121853194409818704428477017060102962218634953171708736539878699060177
7/2
1:
0.1334645495536445136804555518238574810972431255669569336718875587881741342061245437776138728279647818
4
4:
3.399179277799746398292040979199743605908486407406465542716810303626438713067794578393531265695908654
4
2:
0.8572592370087177080280191910455926705100012554361165042099628391452117841434677823902713453550321473
4
8:
5.745719328049896026173387042658107347362337698639997924707211583082976485302837746961257312688295481
4
1:
0.1139289412569228544716196774166261208670219034917146478746111728406160680816031462856436104652857001
9/2
9/2:
6.545473796292022231230584587768254531112267642278113664530843027814559491375737814300651560497735795
9/2
9/4:
1.447696253701786583884030651396968902999805826132653323674426557156968642013490188736000648552878518
9/2
9:
11.22259933874153601243575252411802991184624904105455350328290033144385797394182159599347149242709280
9/2
1:
0.09924648226631347628607066122204031639453980845258143334376965406114797397653609986450128055195709265
5
5:
13.42816115843490212537804002408012235355500109207987095985062096245955317301722628040630115343708073
5
5/2:
2.611727546060370183639684759150672570262163717915288185127074160126749285347916288102856145497823352
5
10:
23.29793548615293425585361280337164536328083467376743931537772132941294494415642951652535332970146349
5
1:
0.08783632385624909629095493950504361602227648293509075699060788966500277658151278178542716751522315647
11/2
11/4:
4.995566845660628013085561739909982087047647365221678163817732419139827809149361372307198382067576649
11/2
11/2:
29.14114650182436985669909493608232729713448065360207769113847516169914269547173157045489846217651822
11/2
11:
51.06701450318509294349824989538775081675672309394279822443687024533257013745047845554420350858462378
11/2
1:
0.07872972902696832169179420533772055632961800700484861553912664157770438714951264603310848813788727317
6
3:
10.07015304375641390161194624466359719720730044796154002667797744581804565171939325984822872551357529
6
6:
66.51844303624665066437746751269957357285619646251960158224280844058483333032886782508303261545235916
6
12:
117.5590764699685954527487620651895685676554233168957139638899485334311382512764688386368440683768465
6
1:
0.07130217810980315985925092736375721266557128364368595044520264662755238716266410556998856323019613860
13/2
13/4:
21.28887352254859012256614809571333758067486870180213972630532213171065318425492145373640095483145571
13/2
13/2:
158.9668895447573820297412643146838090838208205916676251874748951057524534867769090924867183765314372
13/2
13:
282.9909871678189748792467245835510965836307749025906980411491321997621506873318413584839605634893945
13/2
1:
0.06513406847688344770934435919600219236708790749489955095735972697991263357741974982494941041246035866
7
7/2:
47.00742986084665444247497272064086749067528215004122974136383728810782969888896996714594387274471299
7
7:
396.2080397889368303460384360118675157090834210734785661491665284220615914238070472602955241054366861
7
14:
709.7558987921316850954853286376529946353514920067360569927526826745763832874278193869143864676866679
7
1:
0.05993362748737663755998179402108240854761657083034786816337907806785282723108483006278410503525718785
15/2
15/4:
108.0403975330853568145016826546497089819241572740276690260214038280860376872667869705565718253100277
15/2
15/2:
1026.537905580420281636254648526860696324469509032798245890284283563964091023134421542300386415648781
15/2
15:
1848.946155191956172506569970663833049627468719634489636303305842511145106873119110071950769039255533
15/2
1:
0.05549200392830008851521456461255338294026026768507924671500142367197062250832857050502389333507268756
8
4:
257.7134235959505885890497734716186935009115135173083032783805459019553080060101098370745605511586852
8
8:
2757.077520185547793316155867880009628727608563593645536963440841039904410983457134623293594997897733
8
16:
4989.601103996351850483547463853893387604597550354941415365866192228622006204774645898920855770982902
8
1:
0.05165595124019414132434878798611599238750486478066724263581674477750829487269400708234146090088067121
17/2
17/4:
636.3811779185499820053818318855577801802216392467429280334678649869910065650507633266906668611190987
17/2
17/2:
7657.648192188391142233274081283573595226276675824005444665118813187264755618036825033950270080045256
17/2
17:
13916.57268439932329015277253780975000526923330077531119169650104557161112910403049750647336055222293
17/2
1:
0.04831058829080834226858546443268950460614087660632651585467387584231817306756447543053192566712551293
9
9/2:
1623.174839275760476442939539598001854962529702360860003358606744357977686867403550506682495907507633
9
9:
21948.08699372007688386525034929691265128929097059459198056915464552984448260640020196051687012400766
9
18:
40035.50171117306338391224824887472015447559719561773468951546919666877260121720733954780505721803476
9
1:
0.04536816875011080899926653372746707165422778721357010657869715652260486323665225330158441286112572590
19/2
19/4:
4268.096913637224126394196271957985950802032847787364516715383396602098413468589552888597588782904713
19/2
19/2:
64795.35584466540727092618106713893481679764557951524827181823097322482572515263940764829100578191742
19/2
19:
118584.4959459184030483026313449816534162848113097792580736859640552114069409262046532044256671142056
19/2
1:
0.04276055930042858768745267751639992170638632012200755025689114296224297532939823780237409382464721617
10
5:
11549.76543527560208319026403874216836375747201610031484044606357118858848569751639881055891112228591
10
10:
196706.4652124542706469558566703086669736036805749654717792292248181393093363998764356516667688506467
10
20:
361067.2647815613427691332370540245741228965613022957532666642394343415440847861472905540718491181603
10
1:
0.04043407757955495939787503338574277744223895389036312470043760700598227338497047635711244140421188934
Definition
The lower incomplete gamma function is $\gamma(a,x)=\int_0^x t^{a-1}e^{-t}\,\mathrm{d}t$ for $a>0$ and $x>0$ [1], and by analytic continuation in $a$ elsewhere in this table. This table gives its real values at rational $a\notin\mathbb{Z}_{\leq0}$ and rational $x>0$. At $x=0$ and $a>0$ the convergent endpoint value is included.
Parameters
$a$
—   parameter ($a\in\mathbb{Q}\setminus\mathbb{Z}_{\leq0}$)
$x$
—   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 upper incomplete gamma function [1].
(2)
$\gamma(a+1,x)=a\gamma(a,x)-x^ae^{-x}$ [3].
(3)
$\gamma(1/2,x)=\sqrt{\pi}\operatorname{erf}(\sqrt{x})$ [2].
(4)
For nonnegative integers $n$, $\gamma(n+1,x)=n!\left(1-e^{-x}\sum_{k=0}^n x^k/k!\right)$ [2].
(5)
If $a\ne0,-1,-2,\ldots$, then $\gamma(a,x)=x^a M(a;a+1;-x)/a$, where $M$ is Kummer's confluent hypergeometric function [4].
Comments
(6)
For real $a\notin\mathbb{Z}_{\leq0}$ and real $x>0$, the continued value is real. The continuation in $a$ is unique, so no branch has to be chosen.
(7)
The regularized lower incomplete gamma function is $P(a,x)=\gamma(a,x)/\Gamma(a)$ when $\Gamma(a)$ is finite [1]. For positive $a$, $P(a,x)$ is the distribution function of a chi-squared random variable with $2a$ degrees of freedom at $2x$. The regularized value is not stored here.
(8)
The value at $a=0$ is not included because $\gamma(a,x)$ has a pole there. The table of the upper incomplete gamma function stores $\Gamma(0,x)$, which is the exponential integral $E_1(x)$ for positive $x$.
(9)
For a chi-square variable with $n$ degrees of freedom, $\gamma(n/2,x)/\Gamma(n/2)$ is its lower-tail probability at threshold $2x$ [9]. The selected half-integer shape slices compare unit scale, half mean, mean and double mean; only the mean is the intrinsic transition $x=a$, and the other ratios are exploratory reference comparisons. At shape $a=1/2$, $x=k^2/2$ gives the Gaussian integer-standard-deviation family [10]. These are unregularized numerators, not probabilities or quantiles.
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 upper incomplete gamma function $\Gamma(a,x)$ —   for $a\ne0,-1,-2,\ldots$, the lower and upper incomplete gamma functions sum to $\Gamma(a)$
Values of the Gamma function at rational numbers —   contains $\Gamma(a)$, the complete value in the complement formula
Values of the error function $\operatorname{erf}(x)$ —   the row $a=1/2$ is $\sqrt{\pi}\operatorname{erf}(\sqrt{x})$
Values of Kummer's confluent hypergeometric function $M(a;b;z)$ —   $\gamma(a,x)=x^a M(a;a+1;-x)/a$, where the corresponding parameters of $M$ are $(a,a+1,-x)$
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 487 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 $\gamma(1,0)=0$ is included; these bounded families expand both shape and threshold coverage without a Cartesian sampling box)


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