Table of constants with 50 decimal places

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Here are given the first 50 decimal places of constants that occur frequently in numerical computations (see also [1], [2] and [4] for large tables of constants available on the net). This number of digits should be enough for most practical purposes.

1   Related to p


p = 4

n=0 
 (-1)n

2n+1


p
=
3.14159265358979323846264338327950288419716939937510...
p/2
=
1.57079632679489661923132169163975144209858469968755...
p/3
=
1.04719755119659774615421446109316762806572313312503...
p/4
=
0.78539816339744830961566084581987572104929234984377...
p/5
=
0.62831853071795864769252867665590057683943387987502...
p/6
=
0.52359877559829887307710723054658381403286156656251...
2p
=
6.28318530717958647692528676655900576839433879875021...
3p
=
9.42477796076937971538793014983850865259150819812531...
4p
=
12.56637061435917295385057353311801153678867759750042...
p2
=
9.86960440108935861883449099987615113531369940724079...
p3
=
31.00627668029982017547631506710139520222528856588510...
1/p
=
0.31830988618379067153776752674502872406891929148091...
2/p
=
0.63661977236758134307553505349005744813783858296182...


 

p
 
=
1.77245385090551602729816748334114518279754945612238...
1/

 

p
 
=
0.56418958354775628694807945156077258584405062932899...


 

2p
 
=
2.50662827463100050241576528481104525300698674060993...
1/

 

2p
 
=
0.39894228040143267793994605993438186847585863116493...
3
 

p
 
=
1.46459188756152326302014252726379039173859685562793...

2  Related to e


e=

n=0 
 1

n!
.


e
=
2.71828182845904523536028747135266249775724709369995...
e/2
=
1.35914091422952261768014373567633124887862354684997...
2e
=
5.43656365691809047072057494270532499551449418739991...
e2
=
7.38905609893065022723042746057500781318031557055184...
e3
=
20.08553692318766774092852965458171789698790783855415...
1/e
=
0.36787944117144232159552377016146086744581113103176...
e
=
1.64872127070012814684865078781416357165377610071014...
1/e
=
0.60653065971263342360379953499118045344191813548718...
ep
=
23.14069263277926900572908636794854738026610624260021...
e-p
=
0.04321391826377224977441773717172801127572810981063...
ep/2
=
4.81047738096535165547303566670383312639017087466453...
e-p/2
=
0.20787957635076190854695561983497877003387784163176...
eg
=
1.78107241799019798523650410310717954916964521430343...
e-g
=
0.56145948356688516982414321479088078676571038692515...
ee
=
15.15426224147926418976043027262991190552854853685613...
e-e
=
0.065988035845312537076790187596846424938577048252796...

3  Related to Catalan's constant

G is defined by the series expansion
G=

n=0 
 (-1)n

(2n+1)2
.


G
=
0.91596559417721901505460351493238411077414937428167...
1/G
=
1.09174406370390610145415947333389232498605012140824...
G/p
=
0.29156090403081878013838445646839491886406615398583...
p/G
=
3.42981513013245864263455323784799901211670795530093...

4  Square roots


2
=
1.41421356237309504880168872420969807856967187537694...
3
=
1.73205080756887729352744634150587236694280525381038...
5
=
2.23606797749978969640917366873127623544061835961152...
7
=
2.64575131106459059050161575363926042571025918308245...


 

10
 
=
3.16227766016837933199889354443271853371955513932521...
1/2
=
0.70710678118654752440084436210484903928483593768847...
1/3
=
0.57735026918962576450914878050195745564760175127012...
1/5
=
0.44721359549995793928183473374625524708812367192230...
f
=
1.61803398874989484820458683436563811772030917980576...
32
=
1.25992104989487316476721060727822835057025146470150...
42
=
1.18920711500272106671749997056047591529297209246381...
22
=
2.66514414269022518865029724987313984827421131371465...

f = 1/2+5/2 is the golden ratio.

5  Logarithms

We define the logarithm as :
log(x)=
x

1 
 dt

t
.


log(2)
=
0.69314718055994530941723212145817656807550013436025...
log(3)
=
1.09861228866810969139524523692252570464749055782274...
log(5)
=
1.60943791243410037460075933322618763952560135426851...
log(7)
=
1.94591014905531330510535274344317972963708472958186...
log(10)
=
2.30258509299404568401799145468436420760110148862877...
1/log(10)
=
0.43429448190325182765112891891660508229439700580366...
 log(2)

log(3)
=
0.63092975357145743709952711434276085429958564013188...
log( log(2))
=
-0.36651292058166432701243915823266946945426344783711...
log( p)
=
1.14472988584940017414342735135305871164729481291531...
log


 

2p
 

=
0.91893853320467274178032973640561763986139747363778...
log( g)
=
-0.54953931298164482233766176880290778833069898126306...
log( f)
=
0.48121182505960344749775891342436842313518433438566...

6  Gamma and Psi functions

An elementary definition is :


G(x)
=



0 
tx-1e-tdt,       with x > 0,
y(x)
=
 G(x)

G(x)
and we have the important relations
G(1)
=
-g
G(xm)
=
0.


g
=
0.57721566490153286060651209008240243104215933593992...
1/g
=
1.73245471460063347358302531586082968115577655226680...
G(1/2)
=
1.77245385090551602729816748334114518279754945612238...
G(1/3)
=
2.67893853470774763365569294097467764412868937795730...
G(2/3)
=
1.35411793942640041694528802815451378551932726605679...
G(1/4)
=
3.62560990822190831193068515586767200299516768288006...
G(3/4)
=
1.22541670246517764512909830336289052685123924810807...
xm
=
1.46163214496836234126265954232572132846819620400644...
G(xm)
=
0.88560319441088870027881590058258873320795153366990...
y(1/2)
=
-1.96351002602142347944097633299875556719315960466043...
y(1/3)
=
-3.13203378002080632299641907428726885415542829672041...
y(2/3)
=
-1.31823441578658847240234081664511312187136204862767...
y(1/4)
=
-4.22745353337626540808953014609668357736724443870824...
y(3/4)
=
-1.08586087978647216962688676281718069317007503933313...
L
=
5.24411510858423962092967917978223882736550990286325...

L is the Lemniscate constant and its value is given by
L=  G2(1/4)




2p
.

7  Riemann Zeta function

For any s > 1
z(s)=

n=1 
 1

ns
.


z(2)
=
1.64493406684822643647241516664602518921894990120679...
z(3)
=
1.20205690315959428539973816151144999076498629234049...
z(4)
=
1.08232323371113819151600369654116790277475095191872...
z(5)
=
1.03692775514336992633136548645703416805708091950191...
z(6)
=
1.01734306198444913971451792979092052790181749003285...
z(1/2)
=
-1.46035450880958681288949915251529801246722933101258...
z(1/3)
=
-0.97336024835078271546888686244789657077282963174305...
z(2/3)
=
-2.44758073623365823109099570422300521301545223575799...
z(3/2)
=
2.61237534868548834334856756792407163057080065240006...
z(-1/2)
=
-0.20788622497735456601730672539704930222626853128767...
1/z(2)
=
0.607927101854026628663276779258365833426152648033479...
z(0)
=
-0.91893853320467274178032973640561763986139747363778...
z(2)
=
-0.93754825431584375370257409456786497789786028861482...

7.1  Non trivial zeros

First roots of the equation z(s)=0 of the form
s=  1

2
+itm.

t1
=
14.13472514173469379045725198356247027078425711569924...
t2
=
21.02203963877155499262847959389690277733434052490278...
t3
=
25.01085758014568876321379099256282181865954967255799...
t4
=
30.42487612585951321031189753058409132018156002371544...
t5
=
32.93506158773918969066236896407490348881271560351703...
t6
=
37.58617815882567125721776348070533282140559735083079...

You may find very complete tables on the zeros of the Riemann zeta function at [3].

8  Number theory

In this section the letter p always denotes a prime and the sums or products are only involving the set of prime numbers.

References

[1]
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, (1964)

[2]
D.E. Knuth, The Art of Computer Programming, Vol. II, Seminumerical Algorithms, Addison Wesley, (1998)

[3]
A. Odlyzko, Tables of zeros of the Riemann zeta function, http://www.dtc.umn.edu/~odlyzko/

[4]
S. Plouffe, Plouffe's Inverter, http://pi.lacim.uqam.ca/eng/




File translated from TEX by TTH, version 3.01.
On 22 Nov 2002, 14:34.