| L(s) = 1 | − 14·3-s + 14·5-s + 12·7-s − 3.20e3·9-s + 638·11-s − 2.78e3·13-s − 196·15-s + 538·17-s − 4.71e3·19-s − 168·21-s + 4.33e4·23-s − 1.37e5·25-s + 4.07e4·27-s − 1.43e5·29-s + 1.56e5·31-s − 8.93e3·33-s + 168·35-s − 3.34e5·37-s + 3.90e4·39-s − 9.09e5·41-s − 7.74e5·43-s − 4.49e4·45-s + 8.52e5·47-s − 2.05e6·49-s − 7.53e3·51-s − 2.06e6·53-s + 8.93e3·55-s + ⋯ |
| L(s) = 1 | − 0.299·3-s + 0.0500·5-s + 0.0132·7-s − 1.46·9-s + 0.144·11-s − 0.351·13-s − 0.0149·15-s + 0.0265·17-s − 0.157·19-s − 0.00395·21-s + 0.742·23-s − 1.75·25-s + 0.398·27-s − 1.09·29-s + 0.946·31-s − 0.0432·33-s + 0.000662·35-s − 1.08·37-s + 0.105·39-s − 2.06·41-s − 1.48·43-s − 0.0734·45-s + 1.19·47-s − 2.49·49-s − 0.00795·51-s − 1.90·53-s + 0.00723·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2097152 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2097152 ^{s/2} \, \Gamma_{\C}(s+7/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 14 T + 1135 p T^{2} + 5756 p^{2} T^{3} + 1135 p^{8} T^{4} + 14 p^{14} T^{5} + p^{21} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 14 T + 137587 T^{2} - 2578564 p T^{3} + 137587 p^{7} T^{4} - 14 p^{14} T^{5} + p^{21} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 12 T + 293379 p T^{2} + 53820568 T^{3} + 293379 p^{8} T^{4} - 12 p^{14} T^{5} + p^{21} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 58 p T + 17869493 T^{2} - 83450952924 T^{3} + 17869493 p^{7} T^{4} - 58 p^{15} T^{5} + p^{21} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2786 T + 92462683 T^{2} + 176751336556 T^{3} + 92462683 p^{7} T^{4} + 2786 p^{14} T^{5} + p^{21} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 538 T + 599769055 T^{2} - 6045136516972 T^{3} + 599769055 p^{7} T^{4} - 538 p^{14} T^{5} + p^{21} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4718 T + 1418986141 T^{2} - 9609692383652 T^{3} + 1418986141 p^{7} T^{4} + 4718 p^{14} T^{5} + p^{21} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 43316 T + 7071007589 T^{2} - 331921661265816 T^{3} + 7071007589 p^{7} T^{4} - 43316 p^{14} T^{5} + p^{21} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 143674 T + 29082884619 T^{2} + 5591337526936444 T^{3} + 29082884619 p^{7} T^{4} + 143674 p^{14} T^{5} + p^{21} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 156928 T + 39853704861 T^{2} - 10335750019347968 T^{3} + 39853704861 p^{7} T^{4} - 156928 p^{14} T^{5} + p^{21} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 334074 T + 171492687411 T^{2} + 43792698846075740 T^{3} + 171492687411 p^{7} T^{4} + 334074 p^{14} T^{5} + p^{21} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 909698 T + 288771195831 T^{2} + 42998200165964540 T^{3} + 288771195831 p^{7} T^{4} + 909698 p^{14} T^{5} + p^{21} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 774458 T + 883971668661 T^{2} + 419440055538243412 T^{3} + 883971668661 p^{7} T^{4} + 774458 p^{14} T^{5} + p^{21} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 852952 T + 725351339437 T^{2} - 444762288735514960 T^{3} + 725351339437 p^{7} T^{4} - 852952 p^{14} T^{5} + p^{21} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 2063562 T + 3383922054339 T^{2} + 3482640571712643708 T^{3} + 3383922054339 p^{7} T^{4} + 2063562 p^{14} T^{5} + p^{21} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2243922 T + 7938989411301 T^{2} + 11189947819945495172 T^{3} + 7938989411301 p^{7} T^{4} + 2243922 p^{14} T^{5} + p^{21} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 4055170 T + 13121905368395 T^{2} + 25467606155171874412 T^{3} + 13121905368395 p^{7} T^{4} + 4055170 p^{14} T^{5} + p^{21} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 7993670 T + 37636374829229 T^{2} + \)\(11\!\cdots\!12\)\( T^{3} + 37636374829229 p^{7} T^{4} + 7993670 p^{14} T^{5} + p^{21} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 9941468 T + 59931979536597 T^{2} - \)\(21\!\cdots\!16\)\( T^{3} + 59931979536597 p^{7} T^{4} - 9941468 p^{14} T^{5} + p^{21} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 5794470 T + 37828195636791 T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + 37828195636791 p^{7} T^{4} + 5794470 p^{14} T^{5} + p^{21} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1004504 T + 48614168717389 T^{2} - 24671264664594287440 T^{3} + 48614168717389 p^{7} T^{4} - 1004504 p^{14} T^{5} + p^{21} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 14967990 T + 150971280055101 T^{2} + \)\(91\!\cdots\!52\)\( T^{3} + 150971280055101 p^{7} T^{4} + 14967990 p^{14} T^{5} + p^{21} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 4831350 T + 110343597017031 T^{2} + \)\(33\!\cdots\!76\)\( T^{3} + 110343597017031 p^{7} T^{4} + 4831350 p^{14} T^{5} + p^{21} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 5250694 T + 46471338379759 T^{2} + 10300446712993234580 T^{3} + 46471338379759 p^{7} T^{4} + 5250694 p^{14} T^{5} + p^{21} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.34923076889751321252981994862, −10.76577949927302196946806609223, −10.49171085401047291605837754213, −10.15012272475288119897537384969, −9.571142624849671634490475250385, −9.494717913018816881898826886047, −9.030368536072305596390892713401, −8.537723875058443092338755146001, −8.319979329820595772509631031801, −7.966382472392389543750286648056, −7.48227064331914267541638765864, −7.14769758219006927402840942047, −6.62434450812332125749336394743, −6.18447960127659900754494131175, −5.90899671944505553613546437090, −5.60694859708223413030449898304, −4.97138793255569485724551276155, −4.72190096959647931163775117338, −4.31116150975861112408831330514, −3.39406217395737035440565127760, −3.17090207966910544239260136389, −3.01263413532453068529610953143, −1.94996464765087579438251575877, −1.79292360671725755657325669294, −1.24334386638055485701796302455, 0, 0, 0,
1.24334386638055485701796302455, 1.79292360671725755657325669294, 1.94996464765087579438251575877, 3.01263413532453068529610953143, 3.17090207966910544239260136389, 3.39406217395737035440565127760, 4.31116150975861112408831330514, 4.72190096959647931163775117338, 4.97138793255569485724551276155, 5.60694859708223413030449898304, 5.90899671944505553613546437090, 6.18447960127659900754494131175, 6.62434450812332125749336394743, 7.14769758219006927402840942047, 7.48227064331914267541638765864, 7.966382472392389543750286648056, 8.319979329820595772509631031801, 8.537723875058443092338755146001, 9.030368536072305596390892713401, 9.494717913018816881898826886047, 9.571142624849671634490475250385, 10.15012272475288119897537384969, 10.49171085401047291605837754213, 10.76577949927302196946806609223, 11.34923076889751321252981994862
