| L(s) = 1 | − 84·3-s + 1.55e3·5-s − 7.20e3·7-s − 3.89e4·9-s + 3.44e3·11-s − 1.97e4·13-s − 1.30e5·15-s + 1.01e6·17-s − 2.22e5·19-s + 6.05e5·21-s − 1.88e6·23-s − 1.85e5·25-s + 3.84e6·27-s + 4.08e6·29-s − 2.86e6·31-s − 2.89e5·33-s − 1.11e7·35-s + 1.39e6·37-s + 1.66e6·39-s − 1.44e7·41-s + 6.16e7·43-s − 6.06e7·45-s + 1.03e7·47-s + 3.45e7·49-s − 8.54e7·51-s + 6.75e7·53-s + 5.35e6·55-s + ⋯ |
| L(s) = 1 | − 0.598·3-s + 1.11·5-s − 1.13·7-s − 1.98·9-s + 0.0709·11-s − 0.192·13-s − 0.665·15-s + 2.95·17-s − 0.392·19-s + 0.678·21-s − 1.40·23-s − 0.0950·25-s + 1.39·27-s + 1.07·29-s − 0.558·31-s − 0.0424·33-s − 1.26·35-s + 0.122·37-s + 0.115·39-s − 0.796·41-s + 2.74·43-s − 2.20·45-s + 0.309·47-s + 6/7·49-s − 1.76·51-s + 1.17·53-s + 0.0788·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.766466589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.766466589\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 28 p T + 5117 p^{2} T^{2} + 122360 p^{3} T^{3} + 5117 p^{11} T^{4} + 28 p^{19} T^{5} + p^{27} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 1554 T + 520107 p T^{2} - 46663764 p^{2} T^{3} + 520107 p^{10} T^{4} - 1554 p^{18} T^{5} + p^{27} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3444 T + 455343105 T^{2} - 125101303155960 T^{3} + 455343105 p^{9} T^{4} - 3444 p^{18} T^{5} + p^{27} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 19782 T + 18882215055 T^{2} + 378008000651932 T^{3} + 18882215055 p^{9} T^{4} + 19782 p^{18} T^{5} + p^{27} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 1016694 T + 649258140783 T^{2} - 263109059935862868 T^{3} + 649258140783 p^{9} T^{4} - 1016694 p^{18} T^{5} + p^{27} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 222852 T + 614081373717 T^{2} + 186835068238407176 T^{3} + 614081373717 p^{9} T^{4} + 222852 p^{18} T^{5} + p^{27} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 81984 p T + 5417652680517 T^{2} + 5817973976209389696 T^{3} + 5417652680517 p^{9} T^{4} + 81984 p^{19} T^{5} + p^{27} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 4081818 T + 38739015783987 T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + 38739015783987 p^{9} T^{4} - 4081818 p^{18} T^{5} + p^{27} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2869440 T + 21142500166221 T^{2} + \)\(22\!\cdots\!64\)\( T^{3} + 21142500166221 p^{9} T^{4} + 2869440 p^{18} T^{5} + p^{27} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 1395618 T + 262675972194027 T^{2} - \)\(70\!\cdots\!00\)\( T^{3} + 262675972194027 p^{9} T^{4} - 1395618 p^{18} T^{5} + p^{27} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 14420658 T + 764979654799959 T^{2} + \)\(74\!\cdots\!64\)\( T^{3} + 764979654799959 p^{9} T^{4} + 14420658 p^{18} T^{5} + p^{27} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 61631172 T + 2687603003165025 T^{2} - \)\(16\!\cdots\!04\)\( p T^{3} + 2687603003165025 p^{9} T^{4} - 61631172 p^{18} T^{5} + p^{27} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10368960 T + 2946826961339709 T^{2} - \)\(18\!\cdots\!24\)\( T^{3} + 2946826961339709 p^{9} T^{4} - 10368960 p^{18} T^{5} + p^{27} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 67502610 T + 6295046710287531 T^{2} - \)\(20\!\cdots\!32\)\( T^{3} + 6295046710287531 p^{9} T^{4} - 67502610 p^{18} T^{5} + p^{27} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 42590100 T + 19076976504365997 T^{2} - \)\(78\!\cdots\!00\)\( T^{3} + 19076976504365997 p^{9} T^{4} - 42590100 p^{18} T^{5} + p^{27} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 191746842 T + 38596678668907359 T^{2} - \)\(44\!\cdots\!36\)\( T^{3} + 38596678668907359 p^{9} T^{4} - 191746842 p^{18} T^{5} + p^{27} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 255175788 T + 80172518705654361 T^{2} - \)\(11\!\cdots\!08\)\( T^{3} + 80172518705654361 p^{9} T^{4} - 255175788 p^{18} T^{5} + p^{27} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 296514504 T + 147895725194380437 T^{2} + \)\(25\!\cdots\!68\)\( T^{3} + 147895725194380437 p^{9} T^{4} + 296514504 p^{18} T^{5} + p^{27} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 344213310 T + 83298340302082311 T^{2} - \)\(20\!\cdots\!12\)\( T^{3} + 83298340302082311 p^{9} T^{4} - 344213310 p^{18} T^{5} + p^{27} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 960412656 T + 566786434394061357 T^{2} - \)\(21\!\cdots\!28\)\( T^{3} + 566786434394061357 p^{9} T^{4} - 960412656 p^{18} T^{5} + p^{27} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1100517180 T + 873896700882341301 T^{2} - \)\(43\!\cdots\!28\)\( T^{3} + 873896700882341301 p^{9} T^{4} - 1100517180 p^{18} T^{5} + p^{27} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 506816478 T + 956194525794688887 T^{2} - \)\(35\!\cdots\!64\)\( T^{3} + 956194525794688887 p^{9} T^{4} - 506816478 p^{18} T^{5} + p^{27} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 647498250 T + 819696244591424799 T^{2} + \)\(48\!\cdots\!84\)\( T^{3} + 819696244591424799 p^{9} T^{4} + 647498250 p^{18} T^{5} + p^{27} T^{6} \) |
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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
−10.51577814546798453348369595692, −9.774044767825023060481267151363, −9.770164094760371469225003168242, −9.750036193513049679154958771447, −9.013026720279181921712496320317, −8.629485015532953666740537053133, −8.354636365588635050277368087832, −7.70196614638792401387114928651, −7.67247765194062187349982794685, −6.99614571838826159299367744698, −6.33256428097613863551267856096, −6.06837692564788067329713199691, −6.05355998096929881896557211237, −5.48042735980743809131722129479, −5.31084017736087092700882279305, −4.92475904448214918283095815291, −3.85048506004933971653121952462, −3.66372688289524775516944803269, −3.32759639476782011177223208765, −2.49515141905530250310974609915, −2.47917126956516586453061634982, −1.97238671442306773411372940468, −0.931443101815477987428082334087, −0.806361644195198712855562134940, −0.36422511955062043286326279899,
0.36422511955062043286326279899, 0.806361644195198712855562134940, 0.931443101815477987428082334087, 1.97238671442306773411372940468, 2.47917126956516586453061634982, 2.49515141905530250310974609915, 3.32759639476782011177223208765, 3.66372688289524775516944803269, 3.85048506004933971653121952462, 4.92475904448214918283095815291, 5.31084017736087092700882279305, 5.48042735980743809131722129479, 6.05355998096929881896557211237, 6.06837692564788067329713199691, 6.33256428097613863551267856096, 6.99614571838826159299367744698, 7.67247765194062187349982794685, 7.70196614638792401387114928651, 8.354636365588635050277368087832, 8.629485015532953666740537053133, 9.013026720279181921712496320317, 9.750036193513049679154958771447, 9.770164094760371469225003168242, 9.774044767825023060481267151363, 10.51577814546798453348369595692
