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Values of the Scorer function Gi
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Numbers
$x$ 
$\operatorname{Gi}(x)$
0:
0.2049755424820002450503074563645378511982427295495321683469595843380988397685068801764624477817558564
comment: The distinguished origin; see (2).
$a_{1}$:
-0.5784834542745900651035689446248136360258530898589632217294953169791590945514827374510416552598109803
comment: $a_{1}$ is the negative zero of $\operatorname{Ai}$ with index $n=1$, counted from the origin. See (9).
$a_{2}$:
0.3203437313758904012662783589236317561434727335104236154900289332045843959556545722714960492032216100
comment: $a_{2}$ is the negative zero of $\operatorname{Ai}$ with index $n=2$, counted from the origin. See (9).
$a_{3}$:
-0.4249389112286100674873446693417135588517872579330828570536692100933994888570757952027250851154175388
comment: $a_{3}$ is the negative zero of $\operatorname{Ai}$ with index $n=3$, counted from the origin. See (9).
$a_{4}$:
0.3028459394336552231935663873811965900400268232813759101327075025662831954570660292872020711864070960
comment: $a_{4}$ is the negative zero of $\operatorname{Ai}$ with index $n=4$, counted from the origin. See (9).
$a_{5}$:
-0.3759198563004782405220865875695739586267351240498486402300292155928929865047506142529586057164398222
comment: $a_{5}$ is the negative zero of $\operatorname{Ai}$ with index $n=5$, counted from the origin. See (9).
$a_{6}$:
0.2903105652923748798716953327518465295185894601064861642770802636313283624080725020227496517651124119
comment: $a_{6}$ is the negative zero of $\operatorname{Ai}$ with index $n=6$, counted from the origin. See (9).
$a_{7}$:
-0.3485670337037693870839080607648435466475801180651103206495000153437397636153479734237643483682049487
comment: $a_{7}$ is the negative zero of $\operatorname{Ai}$ with index $n=7$, counted from the origin. See (9).
$a_{8}$:
0.2808465633757122435850573764740429304532968205662742135305140674621005237344576027989946329906553344
comment: $a_{8}$ is the negative zero of $\operatorname{Ai}$ with index $n=8$, counted from the origin. See (9).
$a_{9}$:
-0.3301591042951242407641570934908290944338192126013528142885774534764462873091457483220256515004550249
comment: $a_{9}$ is the negative zero of $\operatorname{Ai}$ with index $n=9$, counted from the origin. See (9).
$a_{10}$:
0.2733116120004367131138340417156839239990251962502509985314023009588983708687753625275481610366651688
comment: $a_{10}$ is the negative zero of $\operatorname{Ai}$ with index $n=10$, counted from the origin. See (9).
$a_{11}$:
-0.3165220701223017044234169229360979992110793260548437276790924655357690471354531308784519705016898019
comment: $a_{11}$ is the negative zero of $\operatorname{Ai}$ with index $n=11$, counted from the origin. See (9).
$a_{12}$:
0.2670812097746480085627070918653716340241129536046655712884228031127161065467461109486777789015608361
comment: $a_{12}$ is the negative zero of $\operatorname{Ai}$ with index $n=12$, counted from the origin. See (9).
$a_{13}$:
-0.3058094895925215739379766129666722292728266959494995066925717247520718862958841724991449668104690393
comment: $a_{13}$ is the negative zero of $\operatorname{Ai}$ with index $n=13$, counted from the origin. See (9).
$a_{14}$:
0.2617863400692141218879699152944446828996942011112929076836804674512748978214135066330220381476803601
comment: $a_{14}$ is the negative zero of $\operatorname{Ai}$ with index $n=14$, counted from the origin. See (9).
$a_{15}$:
-0.2970548537054146288789963704369555312495679462072606975126471082787152798219439571660858530041141163
comment: $a_{15}$ is the negative zero of $\operatorname{Ai}$ with index $n=15$, counted from the origin. See (9).
$a_{16}$:
0.2571928899334940142667750423896081642529647858870869103230948017236294843599183073977811381655291202
comment: $a_{16}$ is the negative zero of $\operatorname{Ai}$ with index $n=16$, counted from the origin. See (9).
$a_{17}$:
-0.2896937156331373237883387140808743286515167359971761506176456442293029219246189696008157305121571248
comment: $a_{17}$ is the negative zero of $\operatorname{Ai}$ with index $n=17$, counted from the origin. See (9).
$a_{18}$:
0.2531437001447870048566092914898487869418370163774854561630671601885626991399568979124301062996732408
comment: $a_{18}$ is the negative zero of $\operatorname{Ai}$ with index $n=18$, counted from the origin. See (9).
$a_{19}$:
-0.2833699274325046835088676827855677986702219514584890228448011605347250969327539255721593068964443450
comment: $a_{19}$ is the negative zero of $\operatorname{Ai}$ with index $n=19$, counted from the origin. See (9).
$a_{20}$:
0.2495284436285554861223932827042576083259194709376501859715902284815735828951789583889227369944938911
comment: $a_{20}$ is the negative zero of $\operatorname{Ai}$ with index $n=20$, counted from the origin. See (9).
$\xi_{0}$:
0.2457777895495897625085977944497477339043926604400525347536326958537862419628655855241580223348847167
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{0})=0$; the root convention is (10).
$\xi_{1}$:
-0.5823987884633583494691962382099907162289101700274831026216626412955692276888681705201327453083491588
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{1})=0$; the root convention is (10).
$\xi_{2}$:
0.3203567175655305358698745766402074488746488003276751139485546256503781856592191397164906728720302296
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{2})=0$; the root convention is (10).
$\xi_{3}$:
-0.4251134247033368380863074142013619910170745545328960485524173305166671073626326243079272178488525299
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{3})=0$; the root convention is (10).
$\xi_{4}$:
0.3028537403346461348060731732743482145642804404433940818945961248916184863127398376854710013356716811
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{4})=0$; the root convention is (10).
$\xi_{5}$:
-0.3759650212436919417544686302965199363300140060213764423194251786019239605962954216740588236417958673
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{5})=0$; the root convention is (10).
$\xi_{6}$:
0.2903150589618734615337745520137737126409627816831769429568862819145001130634653422511177329976249675
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{6})=0$; the root convention is (10).
$\xi_{7}$:
-0.3485861137211583246123361742351559905685556381649303258301326090712193484705477170505234425396127329
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{7})=0$; the root convention is (10).
$\xi_{8}$:
0.2808494247417193924302659073497164965238952358357805141658152667590315062558971224363847447223297171
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{8})=0$; the root convention is (10).
$\xi_{9}$:
-0.3301692605403158994254634477109070299995974082558936762527097844000125919532724621609833661067662465
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{9})=0$; the root convention is (10).
$\xi_{10}$:
0.2733135799104154436127654589021028511687385743600706750109112902457417720841480051770745267114579089
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{10})=0$; the root convention is (10).
$\xi_{11}$:
-0.3165282527851176495235736068906334471804973365740129916356812799831015537576905701117771118292567385
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{11})=0$; the root convention is (10).
$\xi_{12}$:
0.2670826413255616986766976447834377522037363386764699310199899755304450140089609948462064723368716231
comment: Extremal height, with $\operatorname{Gi}^{\prime}(\xi_{12})=0$; the root convention is (10).
-1/2:
0.08724360928458474311675656951677720678860953575414109563457920309352730982171551691803521815962225596
comment: A bounded simple-real reference; see (11).
1/2:
0.2447210432765581976910539273564600237077099951710280901146186319457275054818157509942000895212586013
comment: A bounded simple-real reference; see (11).
-1:
-0.1166722172960152826494197605031939643874315735693538981048105202911558154780461724829329581482386393
comment: A bounded simple-real reference; see (11).
1:
0.2352184398104379375986902460186649845769267569597867261803280030100226748152993201979898250945508598
comment: A bounded simple-real reference; see (11).
-2:
-0.5532515841978896890248519050715128238129371267783249157034723910869735887412018554656929462649339153
comment: A bounded simple-real reference; see (11).
2:
0.1689535656540103627737236056565543832351363083601390954033429763326931467230488124515484705542292267
comment: A bounded simple-real reference; see (11).
-4:
0.3146693490272955759649280704033548561046944715372131946949866697816044375854425722786412710519705050
comment: A bounded simple-real reference; see (11).
4:
0.08283430336376872933792404619335125653144212501055055384124138907566439970031465936542965740788094433
comment: A bounded simple-real reference; see (11).
-8:
-0.3708904072242625772850981664528535009830171851997930916382860438752803761743872006435107589986232633
comment: A bounded simple-real reference; see (11).
8:
0.03995113371950890754121959116272127345295274082780519662335612551499510173758101581876417808472075270
comment: A bounded simple-real reference; see (11).
$-3/\sqrt[3]{3}$:
-0.5686334732765030579222695014817154096353043514043368867694288519345349080836892901915078488632006336
comment: A unit-cubic normalization reference; see (12).
$-2/\sqrt[3]{3}$:
-0.3094726274970838905518143846006364389440915917284027130385073310434574598398460466280454260302562155
comment: A unit-cubic normalization reference; see (12).
$-1/\sqrt[3]{3}$:
0.01771456495765806490010658323605824255428416821769360566320751494717493678969279479422600898434526719
comment: A unit-cubic normalization reference; see (12).
$1/\sqrt[3]{3}$:
0.2452031679056050695172067705207961872051505441713737991600963471779796122693272517706021291727968871
comment: A unit-cubic normalization reference; see (12).
$2/\sqrt[3]{3}$:
0.2119037284172500547506767465321969189247571738676473685714871787807051801854853371612945432741545520
comment: A unit-cubic normalization reference; see (12).
$3/\sqrt[3]{3}$:
0.1636332561457967963737280448626201403853058437814380386063650507430009075789434226360317349178665096
comment: A unit-cubic normalization reference; see (12).
Definition
The Scorer function $\operatorname{Gi}$ is defined for real $x$ by the oscillatory improper integral $\operatorname{Gi}(x)=\pi^{-1}\int_0^\infty\sin(xt+t^3/3)\,dt$ [1]. This table gives its values at intrinsic and bounded reference arguments.
Parameters
$x$
—   argument ($x\in\mathbb{R}$)
Formulas
(1)
$\operatorname{Gi}^{\prime\prime}(x)-x\operatorname{Gi}(x)=-1/\pi$ [1].
(2)
With $\Gamma$ denoting the Gamma function, $\operatorname{Gi}(0)=1/(3^{7/6}\Gamma(2/3))$ and $\operatorname{Gi}^{\prime}(0)=1/(3^{5/6}\Gamma(1/3))$ [2].
(3)
For the Airy functions $\operatorname{Ai}$ and $\operatorname{Bi}$, $\operatorname{Gi}(x)=\operatorname{Bi}(x)\left(\frac13-\int_0^x\operatorname{Ai}(t)\,dt\right)+\operatorname{Ai}(x)\int_0^x\operatorname{Bi}(t)\,dt$ [3] [4].
(4)
$\operatorname{Gi}(x)+\operatorname{Hi}(x)=\operatorname{Bi}(x)$ [5].
(5)
$\int_0^\infty\sin(u^3+ku)\,du=\pi3^{-1/3}\operatorname{Gi}(k/3^{1/3})$.
Comments
(6)
The normalization is that of DLMF §9.12 [1]. Derivatives are with respect to $x$, and real powers of $3$ are positive.
(7)
The companion Scorer function $\operatorname{Hi}$ is defined by $\operatorname{Hi}(x)=\pi^{-1}\int_0^\infty e^{xt-t^3/3}\,dt$ [6]. Its relation to $\operatorname{Gi}$ is (4).
(8)
Write $0>a_1>a_2>\cdots$ for the negative zeros of $\operatorname{Ai}$. The label $a_n$ denotes the root itself, not a rounded decimal argument.
(9)
For standard Brownian motion $B$, let $N=\sup_{t\geq0}(B(t)-t^2/2)$. The values $\operatorname{Hi}(a_n)$ occur in series for its distribution, and $\operatorname{Gi}(a_n)$ occur in moment formulas [8]. These applications select Airy zeros rather than a rational sampling grid.
(10)
Let $\xi_0>0$ be the unique positive zero of $\operatorname{Gi}^{\prime}$, and let $0>\xi_1>\xi_2>\cdots$ be its negative zeros [9]. The values here are extremal heights $\operatorname{Gi}(\xi_n)$; the locations are in Real_zeros_of_the_derivative_of_the_Scorer_function_Gi.
(11)
The exploratory reference arguments $x=\pm1/2,\pm1,\pm2,\pm4,\pm8$ give a sparse set of simple real inputs on both sides of the Airy turning point. They complement root-indexed values with near-origin, moderate and larger arguments. This is a bounded identification reference, not a claim that each point has a documented application [10].
(12)
The change of variable $t=3^{1/3}u$ in the defining cubic integrals gives $x=k/3^{1/3}$ when the cubic coefficient is one and the linear coefficient is the integer $k$. The finite range $1\leq|k|\leq3$ is an exploratory simple-coefficient reference choice, not a source-prescribed cutoff [11].
(13)
The current generate.py reproduces the complete attached selection, including the retained entries. Their original written values and entry comments are preserved rather than reassigned to a new source. The selection manifest records the predecessor revision and source hashes; earlier generators and checks remain historical provenance, not the enumeration of this extension.
Programs
(P1)
Python
from generate import generator
from numberdb._write import to_text

for params, result in generator.all_entries():
    print(params["x"], to_text(result["number"], generator.digits_for(params)))
Links
Similar tables
Values of the Airy function of the first kind $\operatorname{Ai}(x)$ —   $\operatorname{Ai}$ occurs in the integral representation (3)
Values of the Airy function of the second kind $\operatorname{Bi}(x)$ —   $\operatorname{Bi}$ is the sum of the two Scorer functions (4)
Values of the Gamma function at rational numbers —   the Gamma values determine $\operatorname{Gi}(0)$ and $\operatorname{Gi}^{\prime}(0)$ (2)
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds the origin and $a_n$ for $1\leq n\leq20$, with $a_n$ defined in (8); these are the arguments in (9), together with $\operatorname{Gi}(\xi_n)$ for $0\leq n\leq12$ under (10); these are finite prefixes of intrinsic indexed families, not a complete real-argument table. In addition, it includes the simple real arguments $\pm1/2,\pm1,\pm2,\pm4,\pm8$ and unit-cubic arguments $k/3^{1/3}$ for $1\leq|k|\leq3$. The exploratory families are identification references with practical finite bounds, not claims of documented applications for every input. All prior entries are retained.)
How they were obtained:

All retained values and metadata are preserved, including their precision. New nonexact values have 100 significant digits. balanced_values.py reproduces the complete selection in balanced-selection.json with Arb balls, including uncertainty in exact irrational arguments and certified roots. A 640-bit guard is supplemented by measured accuracy checks and precision escalation.

more

Every written interval contains a freshly recomputed certified enclosure; every value is also checked with independent mpmath computations at 50 and 100 decimal guard digits. Exact origin anchors stay exact. Scorer series have bounded geometric tails in each residue class. Independent certified Hi quadrature, and Bi minus that integral for Gi, check every value, with a bounded omitted tail and nonzero controls. Gi critical-point ordering and brackets are certified by interval root counting, and their uncertainty is propagated. Structural identities, root endpoint signs where applicable, and selected transformed definite integrals supplement these checks. See balanced-checks-evidence.json. Earlier checks-evidence.json and source attachments describe earlier selections and remain for provenance.



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