# Table of constants with 50 decimal places

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 ¥å 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...
 Ö p
 =
 1.77245385090551602729816748334114518279754945612238...
 1/ Ö p
 =
 0.56418958354775628694807945156077258584405062932899...
 Ö 2p
 =
 2.50662827463100050241576528481104525300698674060993...
 1/ Ö 2p
 =
 0.39894228040143267793994605993438186847585863116493...
 3 Ö p
 =
 1.46459188756152326302014252726379039173859685562793...

## 2  Related to e

 e= ¥å n=0 1 n! .

 e
 =
 2.71828182845904523536028747135266249775724709369995...
 e/2
 =
 1.35914091422952261768014373567633124887862354684997...
 2e
 =
 5.43656365691809047072057494270532499551449418739991...
 e2
 =
 7.38905609893065022723042746057500781318031557055184...
 e3
 =
 20.08553692318766774092852965458171789698790783855415...
 1/e
 =
 0.36787944117144232159552377016146086744581113103176...
 Öe
 =
 1.64872127070012814684865078781416357165377610071014...
 1/Öe
 =
 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= ¥å 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

 Ö2
 =
 1.41421356237309504880168872420969807856967187537694...
 Ö3
 =
 1.73205080756887729352744634150587236694280525381038...
 Ö5
 =
 2.23606797749978969640917366873127623544061835961152...
 Ö7
 =
 2.64575131106459059050161575363926042571025918308245...
 Ö 10
 =
 3.16227766016837933199889354443271853371955513932521...
 1/Ö2
 =
 0.70710678118654752440084436210484903928483593768847...
 1/Ö3
 =
 0.57735026918962576450914878050195745564760175127012...
 1/Ö5
 =
 0.44721359549995793928183473374625524708812367192230...
 f
 =
 1.61803398874989484820458683436563811772030917980576...
 3Ö2
 =
 1.25992104989487316476721060727822835057025146470150...
 4Ö2
 =
 1.18920711500272106671749997056047591529297209246381...
 2Ö2
 =
 2.66514414269022518865029724987313984827421131371465...

f = 1/2+Ö5/2 is the golden ratio.

## 5  Logarithms

We define the logarithm as :
 log(x)= óõ 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 æè Ö 2p öø
 =
 0.91893853320467274178032973640561763986139747363778...
 log( g)
 =
 -0.54953931298164482233766176880290778833069898126306...
 log( f)
 =
 0.48121182505960344749775891342436842313518433438566...

## 6  Gamma and Psi functions

An elementary definition is :

 G(x)
 =
 óõ ¥ 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)

 Ö 2p
.

## 7  Riemann Zeta function

For any s > 1
 z(s)= ¥å 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.

• Mertens constant M
 M
 =
 lim n® ¥ æè å p £ n 1 p -log(log(n)) öø
 M
 =
 0.26149721284764278375542683860869585905156664826119...
• Artin's constant
 C
 =
 Õ p ³ 2 æè 1- 1 p(p-1) öø
 C
 =
 0.37395581361920228805472805434641641511162924860615...
• Twin prime constant
 C2
 =
 Õ p ³ 3 æè 1- 1 (p-1)2 öø
 C2
 =
 0.66016181584686957392781211001455577843262336028473...
 2C2
 =
 1.32032363169373914785562422002911155686524672056946...
• Landau-Ramanujan constant
 K=0.76422365358922066299069873125009232811679054139340...

## 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.