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Hardy-Littlewood singular series of prime tuples
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Numbers
$H$ 
$\mathfrak S(H)$
0, 2:
1.320323631693739147855624220029111556865246720569466826638896846670811284608990554287520062827679736
comment: The pair offset is $D=2$; this is $2C_2$, twice the twin prime constant $C_2$.
0, 4:
1.320323631693739147855624220029111556865246720569466826638896846670811284608990554287520062827679736
comment: The pair offset is $D=4$; this is $2C_2$, twice the twin prime constant $C_2$.
0, 6:
2.640647263387478295711248440058223113730493441138933653277793693341622569217981108575040125655359472
comment: The pair offset is $D=6$; this is $2\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 8:
1.320323631693739147855624220029111556865246720569466826638896846670811284608990554287520062827679736
comment: The pair offset is $D=8$; this is $2C_2$, twice the twin prime constant $C_2$.
0, 10:
1.760431508924985530474165626705482075820328960759289102185195795561081712811987405716693417103572981
comment: The pair offset is $D=10$; this is $\frac{4}{3}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 12:
2.640647263387478295711248440058223113730493441138933653277793693341622569217981108575040125655359472
comment: The pair offset is $D=12$; this is $2\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 14:
1.584388358032486977426749064034933868238296064683360191966676216004973541530788665145024075393215683
comment: The pair offset is $D=14$; this is $\frac{6}{5}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 16:
1.320323631693739147855624220029111556865246720569466826638896846670811284608990554287520062827679736
comment: The pair offset is $D=16$; this is $2C_2$, twice the twin prime constant $C_2$.
0, 18:
2.640647263387478295711248440058223113730493441138933653277793693341622569217981108575040125655359472
comment: The pair offset is $D=18$; this is $2\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 20:
1.760431508924985530474165626705482075820328960759289102185195795561081712811987405716693417103572981
comment: The pair offset is $D=20$; this is $\frac{4}{3}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 22:
1.467026257437487942061804688921235063183607467299407585154329829634234760676656171430577847586310818
comment: The pair offset is $D=22$; this is $\frac{10}{9}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 24:
2.640647263387478295711248440058223113730493441138933653277793693341622569217981108575040125655359472
comment: The pair offset is $D=24$; this is $2\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 26:
1.440353052756806343115226421849939880216632786075781992696978378186339583209807877404567341266559712
comment: The pair offset is $D=26$; this is $\frac{12}{11}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 28:
1.584388358032486977426749064034933868238296064683360191966676216004973541530788665145024075393215683
comment: The pair offset is $D=28$; this is $\frac{6}{5}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 30:
3.520863017849971060948331253410964151640657921518578204370391591122163425623974811433386834207145962
comment: The pair offset is $D=30$; this is $\frac{8}{3}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 32:
1.320323631693739147855624220029111556865246720569466826638896846670811284608990554287520062827679736
comment: The pair offset is $D=32$; this is $2C_2$, twice the twin prime constant $C_2$.
0, 34:
1.408345207139988424379332501364385660656263168607431281748156636448865370249589924573354733682858385
comment: The pair offset is $D=34$; this is $\frac{16}{15}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 36:
2.640647263387478295711248440058223113730493441138933653277793693341622569217981108575040125655359472
comment: The pair offset is $D=36$; this is $2\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 38:
1.397989727675723803611837409442588707269084762955906051735302543533800183703637057480903595935190309
comment: The pair offset is $D=38$; this is $\frac{18}{17}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 40:
1.760431508924985530474165626705482075820328960759289102185195795561081712811987405716693417103572981
comment: The pair offset is $D=40$; this is $\frac{4}{3}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 42:
3.168776716064973954853498128069867736476592129366720383933352432009947083061577330290048150786431366
comment: The pair offset is $D=42$; this is $\frac{12}{5}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 44:
1.467026257437487942061804688921235063183607467299407585154329829634234760676656171430577847586310818
comment: The pair offset is $D=44$; this is $\frac{10}{9}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 46:
1.383196185583917202515415849554307345287401326310870008859796696512278488637990104491687684867093057
comment: The pair offset is $D=46$; this is $\frac{22}{21}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 48:
2.640647263387478295711248440058223113730493441138933653277793693341622569217981108575040125655359472
comment: The pair offset is $D=48$; this is $2\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 50:
1.760431508924985530474165626705482075820328960759289102185195795561081712811987405716693417103572981
comment: The pair offset is $D=50$; this is $\frac{4}{3}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 52:
1.440353052756806343115226421849939880216632786075781992696978378186339583209807877404567341266559712
comment: The pair offset is $D=52$; this is $\frac{12}{11}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 54:
2.640647263387478295711248440058223113730493441138933653277793693341622569217981108575040125655359472
comment: The pair offset is $D=54$; this is $2\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 56:
1.584388358032486977426749064034933868238296064683360191966676216004973541530788665145024075393215683
comment: The pair offset is $D=56$; this is $\frac{6}{5}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 58:
1.369224506941655412591017709659819392304700302812780412810707840991952443298212426668539324413890096
comment: The pair offset is $D=58$; this is $\frac{28}{27}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 60:
3.520863017849971060948331253410964151640657921518578204370391591122163425623974811433386834207145962
comment: The pair offset is $D=60$; this is $\frac{8}{3}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 62:
1.365852032786626704678231951754253334688186262658069131005755358624977190974817814780193168442427313
comment: The pair offset is $D=62$; this is $\frac{30}{29}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 64:
1.320323631693739147855624220029111556865246720569466826638896846670811284608990554287520062827679736
comment: The pair offset is $D=64$; this is $2C_2$, twice the twin prime constant $C_2$.
0, 66:
2.934052514874975884123609377842470126367214934598815170308659659268469521353312342861155695172621635
comment: The pair offset is $D=66$; this is $\frac{20}{9}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 68:
1.408345207139988424379332501364385660656263168607431281748156636448865370249589924573354733682858385
comment: The pair offset is $D=68$; this is $\frac{16}{15}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 70:
2.112517810709982636568998752046578490984394752911146922622234954673298055374384886860032100524287577
comment: The pair offset is $D=70$; this is $\frac{8}{5}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 72:
2.640647263387478295711248440058223113730493441138933653277793693341622569217981108575040125655359472
comment: The pair offset is $D=72$; this is $2\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 74:
1.358047164027845980651499197744229029918539484014308735971436756575691607026390284410020636051327728
comment: The pair offset is $D=74$; this is $\frac{36}{35}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 76:
1.397989727675723803611837409442588707269084762955906051735302543533800183703637057480903595935190309
comment: The pair offset is $D=76$; this is $\frac{18}{17}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 78:
2.880706105513612686230452843699879760433265572151563985393956756372679166419615754809134682533119424
comment: The pair offset is $D=78$; this is $\frac{24}{11}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 80:
1.760431508924985530474165626705482075820328960759289102185195795561081712811987405716693417103572981
comment: The pair offset is $D=80$; this is $\frac{4}{3}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 82:
1.354178083788450408057050482081140058323329969814837770911689073508524394470759542858994936233517678
comment: The pair offset is $D=82$; this is $\frac{40}{39}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 84:
3.168776716064973954853498128069867736476592129366720383933352432009947083061577330290048150786431366
comment: The pair offset is $D=84$; this is $\frac{12}{5}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 86:
1.352526647100903517315517493688358180203423469851648944361796769760343267160429348294532747286891437
comment: The pair offset is $D=86$; this is $\frac{42}{41}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 88:
1.467026257437487942061804688921235063183607467299407585154329829634234760676656171430577847586310818
comment: The pair offset is $D=88$; this is $\frac{10}{9}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 90:
3.520863017849971060948331253410964151640657921518578204370391591122163425623974811433386834207145962
comment: The pair offset is $D=90$; this is $\frac{8}{3}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 92:
1.383196185583917202515415849554307345287401326310870008859796696512278488637990104491687684867093057
comment: The pair offset is $D=92$; this is $\frac{22}{21}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 94:
1.349664156842488906696860313807536258128918869915454978341983443263495979822523677716131619779405952
comment: The pair offset is $D=94$; this is $\frac{46}{45}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 96:
2.640647263387478295711248440058223113730493441138933653277793693341622569217981108575040125655359472
comment: The pair offset is $D=96$; this is $2\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 98:
1.584388358032486977426749064034933868238296064683360191966676216004973541530788665145024075393215683
comment: The pair offset is $D=98$; this is $\frac{6}{5}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 100:
1.760431508924985530474165626705482075820328960759289102185195795561081712811987405716693417103572981
comment: The pair offset is $D=100$; this is $\frac{4}{3}\cdot2C_2$, where $C_2$ is the twin prime constant.
0, 2, 6:
2.858248595719220432430134660726350878039295592995676029048805072190530759022626346936131124043820779
comment: This is a prime triplet; the finite factor before the generic tail over primes $p>6$ is $\frac{225}{64}$; OEIS A271886 stores this full normalization; A065418 omits the factor $9/2$.
0, 4, 6:
2.858248595719220432430134660726350878039295592995676029048805072190530759022626346936131124043820779
comment: This is a prime triplet; the finite factor before the generic tail over primes $p>6$ is $\frac{225}{64}$; OEIS A271886 stores this full normalization; A065418 omits the factor $9/2$.
0, 2, 6, 8:
4.151180863237415757165285561959537515799410019333963032027163349521998358505355429986843573203151668
comment: This is a prime quadruplet; the finite factor before the generic tail over primes $p>8$ is $\frac{42875}{8192}$; OEIS A061642 stores this full normalization; A065419 omits the factor $27/2$.
0, 2, 6, 8, 12:
10.13179494999607984398842718486766891772613497120684297361244149921707370913429052048440088332191249
comment: This is a prime quintuplet; the finite factor before the generic tail over primes $p>12$ is $\frac{35153041}{2621440}$; OEIS A269843 stores only the generic tail, so the full singular series multiplies it by $50625/2048$.
0, 4, 6, 10, 12:
10.13179494999607984398842718486766891772613497120684297361244149921707370913429052048440088332191249
comment: This is a prime quintuplet; the finite factor before the generic tail over primes $p>12$ is $\frac{35153041}{2621440}$; OEIS A269843 stores only the generic tail, so the full singular series multiplies it by $50625/2048$.
0, 4, 6, 10, 12, 16:
17.29861231158488860612210771577368396573561425086143534997161335635083046593648497366951615332486476
comment: This is a prime sextuplet; the finite factor before the generic tail over primes $p>16$ is $\frac{7035070070035007}{300578991243264}$.
0, 2, 6, 8, 12, 18, 20:
53.97194830012965239607302910617223899785127235775354158588040176132398453301669745930574981427398218
comment: This is a prime 7-tuplet; the finite factor before the generic tail over primes $p>20$ is $\frac{1142403694322595171957595353537289}{16465873788275112647519524356096}$.
0, 2, 8, 12, 14, 18, 20:
53.97194830012965239607302910617223899785127235775354158588040176132398453301669745930574981427398218
comment: This is a prime 7-tuplet; the finite factor before the generic tail over primes $p>20$ is $\frac{1142403694322595171957595353537289}{16465873788275112647519524356096}$.
0, 2, 6, 8, 12, 18, 20, 26:
178.2619543965424453953607880821794581469210008901717741431816754116260050337825547796611668894947829
comment: This is a prime 8-tuplet; the finite factor before the generic tail over primes $p>26$ is $\frac{64536031317917576930966860938199412058679}{276251521174629832311766924339483508736}$.
0, 2, 6, 12, 14, 20, 24, 26:
475.3652117241131877209621015524785550584560023737913977151511344310026800900868127457631117053194212
comment: This is a prime 8-tuplet; the finite factor before the generic tail over primes $p>26$ is $\frac{64536031317917576930966860938199412058679}{103594320440486187116912596627306315776}$.
0, 6, 8, 14, 18, 20, 24, 26:
178.2619543965424453953607880821794581469210008901717741431816754116260050337825547796611668894947829
comment: This is a prime 8-tuplet; the finite factor before the generic tail over primes $p>26$ is $\frac{64536031317917576930966860938199412058679}{276251521174629832311766924339483508736}$.
0, 2, 6, 8, 12, 18, 20, 26, 30:
630.0643626548097478661627389895866678482559592883904306575109476566857524506346971351807676690153157
comment: This is a prime 9-tuplet; the finite factor before the generic tail over primes $p>30$ is $\frac{94648487250960773126134150403729770271603352319236025}{111497734277949416291473115453964184520848815685632}$.
0, 4, 6, 10, 16, 18, 24, 28, 30:
1260.128725309619495732325477979173335696511918576780861315021895313371504901269394270361535338030631
comment: This is a prime 9-tuplet; the finite factor before the generic tail over primes $p>30$ is $\frac{94648487250960773126134150403729770271603352319236025}{55748867138974708145736557726982092260424407842816}$.
0, 2, 6, 12, 14, 20, 24, 26, 30:
1260.128725309619495732325477979173335696511918576780861315021895313371504901269394270361535338030631
comment: This is a prime 9-tuplet; the finite factor before the generic tail over primes $p>30$ is $\frac{94648487250960773126134150403729770271603352319236025}{55748867138974708145736557726982092260424407842816}$.
0, 4, 10, 12, 18, 22, 24, 28, 30:
630.0643626548097478661627389895866678482559592883904306575109476566857524506346971351807676690153157
comment: This is a prime 9-tuplet; the finite factor before the generic tail over primes $p>30$ is $\frac{94648487250960773126134150403729770271603352319236025}{111497734277949416291473115453964184520848815685632}$.
0, 2, 6, 8, 12, 18, 20, 26, 30, 32:
1704.740940196652853231030001915991744701693413308015085568849452449905813824924894494420306959332636
comment: This is a prime 10-tuplet [14]; the finite factor in (3) is included in the value.
0, 2, 6, 12, 14, 20, 24, 26, 30, 32:
1704.740940196652853231030001915991744701693413308015085568849452449905813824924894494420306959332636
comment: This is a prime 10-tuplet [15]; the finite factor in (3) is included in the value.
0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36:
3062.079326363500957396951485599341286683184749054066788855350409975192900874043313016182662451878442
comment: This is a prime 11-tuplet [16]; the finite factor in (3) is included in the value.
0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36:
3062.079326363500957396951485599341286683184749054066788855350409975192900874043313016182662451878442
comment: This is a prime 11-tuplet [17]; the finite factor in (3) is included in the value.
0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42:
9931.315645884895529683039542085719579614038818280191980092243690131411464908285409572721806171356295
comment: This is a prime 12-tuplet [19]; the finite factor in (3) is included in the value.
0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42:
9931.315645884895529683039542085719579614038818280191980092243690131411464908285409572721806171356295
comment: This is a prime 12-tuplet [18]; the finite factor in (3) is included in the value.
Definition
For an admissible prime $k$-tuple with offsets $H=(0,h_2,\ldots,h_k)$ [3], ordered by $0<h_2<\cdots<h_k$, this table stores the Hardy-Littlewood singular series $\mathfrak S(H)$ [4] [5] in the full Euler-product normalization of (1).
Parameters
$H$
—   offset set of the prime tuple ($H=(0,h_2,\ldots,h_k)$ with $0<h_2<\cdots<h_k$, all $h_i$ integers, and $H$ admissible)
Formulas
(1)
$\mathfrak S(H)=\prod_p\frac{1-w_H(p)/p}{(1-1/p)^k}$, where $w_H(p)=|\{h\bmod p:h\in H\}|$. The tuple is admissible exactly when $w_H(p)<p$ for every prime $p$.
(2)
The first Hardy-Littlewood conjecture predicts $\#\{n\leq x:n+h\text{ is prime for all }h\in H\}\sim \mathfrak S(H)\int_2^x\frac{dt}{(\log t)^k}$ [1] [4].
(3)
If $d=\max H$, then $\mathfrak S(H)=\left(\prod_{p\leq d}\frac{1-w_H(p)/p}{(1-1/p)^k}\right) \prod_{p>d}\frac{p^{k-1}(p-k)}{(p-1)^k}$, because for $p>d$ all offsets in $H$ are distinct modulo $p$.
(4)
For even $D$, $\mathfrak S((0,D))=2C_2\prod_{\substack{p\mid D\\p>2}}\frac{p-1}{p-2}$, where $C_2$ is the twin prime constant.
(5)
$\mathfrak S((0,2,6))=\mathfrak S((0,4,6))=\frac92 \prod_{p\geq5}\frac{p^2(p-3)}{(p-1)^3}$ [9], and $\mathfrak S((0,2,6,8))=\frac{27}{2} \prod_{p\geq5}\frac{p^3(p-4)}{(p-1)^4}$ [11].
Comments
(6)
The normalization is the one in (1). Thus $\mathfrak S((0,2))=2C_2$, twice the twin prime constant $C_2$. Some OEIS entries for longer tuples store only a generic Euler-product tail; the entries here include every finite local factor in (3).
(7)
The table holds every pair $H=(0,D)$ with $D$ even and $2\leq D\leq 100$, and every prime constellation with $3\leq k\leq 12$, that is, every admissible $H$ of least diameter for its $k$ [3] [6]. It does not hold all admissible offset sets of bounded diameter.
(8)
By (1), $\mathfrak S(H)$ depends on $H$ only through the residue counts $w_H(p)$. Reflecting a tuple of diameter $d$ by $h\mapsto d-h$ only translates and negates residue classes modulo each prime $p$, so mirror-image constellations have the same value. For pairs $H=(0,D)$, (4) shows that the value depends only on the odd prime divisors of $D$.
(9)
The product defining $\mathfrak S(H)$ converges for every admissible offset set $H$, and the Hardy-Littlewood asymptotic that gives it as a leading constant remains conjectural. The table records constants of the conjecture, not counts of proved infinite families of primes.
(10)
The OEIS entries A065418, A065419 and A269843 record generic tails for the triplet, quadruplet and quintuplet products [10] [12] [13]. For the quintuplet rows, the factor missing from A269843's $p>5$ tail is $50625/2048$, so the entries here are $50625/2048$ times that tail.
(11)
Asked for in [2], as Hardy-Littlewood singular series for small $D$.
Programs
(P1)
Sage
from sage.libs.pari import pari
H = [0, 2, 6, 8, 12]
k = len(H)
start = max(H) + 1
def local(p):
    w = len({h % p for h in H})
    return (1 - QQ(w)/p) / (1 - QQ(1)/p)^k
small = prod(local(p) for p in prime_range(2, start))
tail = pari('prodeulerrat(p^%d*(p-%d)/(p-1)^%d, 1, %d)' % (k-1, k, k, start))
pari(str(small)) * tail       # 10.1317949499960798439884271848...
(P2)
PARI/GP
default(realprecision, 120)
2 * prodeulerrat(1 - 1/(p-1)^2, 1, 3)                 \\ H = (0,2)
(9/2) * prodeulerrat(p^2*(p-3)/(p-1)^3, 1, 5)          \\ H = (0,2,6)
(27/2) * prodeulerrat(p^3*(p-4)/(p-1)^4, 1, 5)         \\ H = (0,2,6,8)
References
[1]
G. H. Hardy and J. E. Littlewood, Some problems of Partitio numerorum; III: On the expression of a number as a sum of primes, Acta Mathematica 44 (1923), 1-70.
[2]
numberdb-data issue #131, "Hardy-Littlewood singular series for small D", https://github.com/numberdb/numberdb-data/issues/131
Links
Similar tables
Twin prime constant —   $C_2$ is half the entry $\mathfrak S((0,2))$ in this normalization
Values of the prime zeta function at rational numbers —   the same prime-indexed Euler products can be expanded as convergent combinations of prime zeta values
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds every pair $H=(0,D)$ with $D$ even and $2\leq D\leq 100$, and every prime constellation with $3\leq k\leq 12$)
How they were obtained:

The finite part of (3) is computed exactly in $\mathbb{Q}$ from the residue classes of $H$ modulo each prime $p\leq \max H$. The generic tail is computed with PARI's prodeulerrat [7] at $140$ decimal digits, and $100$ digits are written.

more

Before the draft was filled, the pair formula (4) was tested on every pair in the table; the triplet, quadruplet and quintuplet normalizations were compared with the corresponding OEIS entries and tail entries; the full Euler product truncated at a large prime was used as an independent convergence control; and the program snippets were run on the displayed entries.



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