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Values of the Scorer function Hi
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Numbers
$x$ 
$\operatorname{Hi}(x)$
0:
0.4099510849640004901006149127290757023964854590990643366939191686761976795370137603529248955635117128
comment: The distinguished origin; see (2).
$a_{1}$:
0.1245402522162542815281149909714380315264394596838000730752416184399647527241647746013012486737100440
comment: $a_{1}$ is the negative zero of $\operatorname{Ai}$ with index $n=1$, counted from the origin. See (8).
$a_{2}$:
0.07600215315639298448020071727434630049779472380132126640918992141838646420562093879629301134613907004
comment: $a_{2}$ is the negative zero of $\operatorname{Ai}$ with index $n=2$, counted from the origin. See (8).
$a_{3}$:
0.05703737973165383593296271021685351061035745077455567972333009168707879348878135897273507486918615981
comment: $a_{3}$ is the negative zero of $\operatorname{Ai}$ with index $n=3$, counted from the origin. See (8).
$a_{4}$:
0.04661843846743726753344172842478982875640137685432877826829230965350319293673178204506560041679312460
comment: $a_{4}$ is the negative zero of $\operatorname{Ai}$ with index $n=4$, counted from the origin. See (8).
$a_{5}$:
0.03991448564742125669566930165836905255961731102835035231697950446873818246851277710015607185586480642
comment: $a_{5}$ is the negative zero of $\operatorname{Ai}$ with index $n=5$, counted from the origin. See (8).
$a_{6}$:
0.03518535667043631586589037448426885568432952926363579789412959087373201483572904031521366037318249424
comment: $a_{6}$ is the negative zero of $\operatorname{Ai}$ with index $n=6$, counted from the origin. See (8).
$a_{7}$:
0.03164214814398032708253598192720720533644628779679680653720382085609845112726413688138720109455084960
comment: $a_{7}$ is the negative zero of $\operatorname{Ai}$ with index $n=7$, counted from the origin. See (8).
$a_{8}$:
0.02887213228710768858135016413434382849343783295375526493665323761172442589889857209004362950227201580
comment: $a_{8}$ is the negative zero of $\operatorname{Ai}$ with index $n=8$, counted from the origin. See (8).
$a_{9}$:
0.02663701134090775346903644565810503585083163743072556882325888863404854583035991276419760701227809679
comment: $a_{9}$ is the negative zero of $\operatorname{Ai}$ with index $n=9$, counted from the origin. See (8).
$a_{10}$:
0.02478889068925084285367864243809333091011985050588270318365545977704393112889266885838458160054723085
comment: $a_{10}$ is the negative zero of $\operatorname{Ai}$ with index $n=10$, counted from the origin. See (8).
$a_{11}$:
0.02323076285864876304876209878253505083720946807274637984789800543520500026110428026633223593323323256
comment: $a_{11}$ is the negative zero of $\operatorname{Ai}$ with index $n=11$, counted from the origin. See (8).
$a_{12}$:
0.02189615334568872738947351332760099673166349977391397169153901399330615992141213473361256912480101271
comment: $a_{12}$ is the negative zero of $\operatorname{Ai}$ with index $n=12$, counted from the origin. See (8).
$a_{13}$:
0.02073786524473654067556711650835895970222315729882407300610658344965976775918057845972915694256169480
comment: $a_{13}$ is the negative zero of $\operatorname{Ai}$ with index $n=13$, counted from the origin. See (8).
$a_{14}$:
0.01972139012464092769077029656821583219822870607562060084760004126805221819921188804717495470831105463
comment: $a_{14}$ is the negative zero of $\operatorname{Ai}$ with index $n=14$, counted from the origin. See (8).
$a_{15}$:
0.01882086556500208880841451248303655533455309379411736971016168788101838139194163316817624666606680203
comment: $a_{15}$ is the negative zero of $\operatorname{Ai}$ with index $n=15$, counted from the origin. See (8).
$a_{16}$:
0.01801649586494108259295968112621570388398613297033673949088557265968525714106890759901085732064868298
comment: $a_{16}$ is the negative zero of $\operatorname{Ai}$ with index $n=16$, counted from the origin. See (8).
$a_{17}$:
0.01729285054174053292091093660294514418172443798706095917178428222260110353239295666992675896273441729
comment: $a_{17}$ is the negative zero of $\operatorname{Ai}$ with index $n=17$, counted from the origin. See (8).
$a_{18}$:
0.01663770921751606613932403969207029914039785912107831141154280765873492214647142794837928976080519425
comment: $a_{18}$ is the negative zero of $\operatorname{Ai}$ with index $n=18$, counted from the origin. See (8).
$a_{19}$:
0.01604125761704278585349603790807061741490907674779142731078262376733438325500531005078584300588831463
comment: $a_{19}$ is the negative zero of $\operatorname{Ai}$ with index $n=19$, counted from the origin. See (8).
$a_{20}$:
0.01549551555089711145581030452163063480213905510847780867343287067402427060162611548230814515832204210
comment: $a_{20}$ is the negative zero of $\operatorname{Ai}$ with index $n=20$, counted from the origin. See (8).
-1/2:
0.2931090504664691070529559242094872958823166492593338803489597265607978326948761743238613079352170332
comment: A bounded simple-real reference; see (9).
1/2:
0.6095559998265972956089948714387831571490774053337115368193901096056810591475095145648228585430459362
comment: A bounded simple-real reference; see (9).
-1:
0.2206696067929598945381097514817931090244496849618778705400674935257939738562069297066387582017714815
comment: A bounded simple-real reference; see (9).
1:
0.9722051551424333218376885710096220108084221876846558113582929686715946742112511629144829985557544346
comment: A bounded simple-real reference; see (9).
-2:
0.1409489962414912009416178504600517817784022923059201868795954268115939270658205792954676924067579409
comment: A bounded simple-real reference; see (9).
2:
3.129141434324204347506880819566898038768839655043481692279580568504466000587936816798803732502712418
comment: A bounded simple-real reference; see (9).
-4:
0.07756535667970371358956375724348862713691770079567790490036650348981801415579445118638219138326172747
comment: A bounded simple-real reference; see (9).
4:
83.76423710510437119324256641485292498408037736001360506393336149367129436340697978100695226144151348
comment: A bounded simple-real reference; see (9).
-8:
0.03963882647312471731522192717714648117058274026910091421286725470465692892666851607406206744781245149
comment: A bounded simple-real reference; see (9).
8:
1199585.964173326211372746958378829736857509674757436995523581517875592740645472809829365730586798311
comment: A bounded simple-real reference; see (9).
$-3/\sqrt[3]{3}$:
0.1367292036991777385883903784787914237532082538535511873892994084398840627396813561390559571525044483
comment: A unit-cubic normalization reference; see (10).
$-2/\sqrt[3]{3}$:
0.1824877353510013066864215088639669055900172437664518812665916920843943748292287649397961690892825724
comment: A unit-cubic normalization reference; see (10).
$-1/\sqrt[3]{3}$:
0.2611672954567928556239394606558660197708905551003650272233975899287252668066513764336013599034418960
comment: A unit-cubic normalization reference; see (10).
$1/\sqrt[3]{3}$:
0.7237960109743029191642407637839780475814018095032649635099598256772351706412169260560330333158052380
comment: A unit-cubic normalization reference; see (10).
$2/\sqrt[3]{3}$:
1.470367027359469536623636669549755863812904152259439350322352901081610707459608390470527971681631640
comment: A unit-cubic normalization reference; see (10).
$3/\sqrt[3]{3}$:
3.485034351790325789338680371930290714610233208462100560025133352503327396816155146245271885974606354
comment: A unit-cubic normalization reference; see (10).
Definition
The Scorer function $\operatorname{Hi}$ is defined for real $x$ by $\operatorname{Hi}(x)=\pi^{-1}\int_0^\infty e^{xt-t^3/3}\,dt$ [1]. This table gives its values at intrinsic and bounded reference arguments.
Parameters
$x$
—   argument ($x\in\mathbb{R}$)
Formulas
(1)
$\operatorname{Hi}^{\prime\prime}(x)-x\operatorname{Hi}(x)=1/\pi$ [1].
(2)
With $\Gamma$ denoting the Gamma function, $\operatorname{Hi}(0)=2/(3^{7/6}\Gamma(2/3))$ and $\operatorname{Hi}^{\prime}(0)=2/(3^{5/6}\Gamma(1/3))$ [2].
(3)
For the companion Scorer function $\operatorname{Gi}$ and the Airy function $\operatorname{Bi}$, $\operatorname{Gi}(x)+\operatorname{Hi}(x)=\operatorname{Bi}(x)$ [3].
(4)
$\int_0^\infty e^{-u^3+ku}\,du=\pi3^{-1/3}\operatorname{Hi}(k/3^{1/3})$.
Comments
(5)
The normalization is that of DLMF §9.12 [1]. There is no scaling by $\pi$ or subtraction of a homogeneous Airy solution. Derivatives are with respect to $x$, and real powers of $3$ are positive.
(6)
For every real $x$, $\operatorname{Hi}(x)>0$ and $\operatorname{Hi}^{\prime}(x)>0$. The integrand in the definition is positive, and differentiation under the integral sign introduces the positive factor $t$.
(7)
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.
(8)
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 [6]. These applications select Airy zeros rather than a rational sampling grid.
(9)
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 [8].
(10)
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 [9].
(11)
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 Scorer function Gi —   the companion function, sharing the origin and Airy-zero arguments and satisfying (3)
Values of the Airy function of the second kind $\operatorname{Bi}(x)$ —   $\operatorname{Bi}$ is the sum of the two Scorer functions (3)
Values of the Gamma function at rational numbers —   the Gamma values determine $\operatorname{Hi}(0)$ and $\operatorname{Hi}^{\prime}(0)$ (2)
Data properties
Entries are of type: real number
Table is complete: no (it holds the origin and $a_n$ for $1\leq n\leq20$, with $a_n$ defined in (7); these are the arguments in (8); 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 well the digits are known: proven
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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