| L(s) = 1 | − 6·2-s + 9·3-s + 24·4-s + 10·5-s − 54·6-s + 7·7-s − 80·8-s + 54·9-s − 60·10-s − 44·11-s + 216·12-s + 9·13-s − 42·14-s + 90·15-s + 240·16-s − 84·17-s − 324·18-s + 240·20-s + 63·21-s + 264·22-s − 2·23-s − 720·24-s − 179·25-s − 54·26-s + 270·27-s + 168·28-s − 92·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s + 0.894·5-s − 3.67·6-s + 0.377·7-s − 3.53·8-s + 2·9-s − 1.89·10-s − 1.20·11-s + 5.19·12-s + 0.192·13-s − 0.801·14-s + 1.54·15-s + 15/4·16-s − 1.19·17-s − 4.24·18-s + 2.68·20-s + 0.654·21-s + 2.55·22-s − 0.0181·23-s − 6.12·24-s − 1.43·25-s − 0.407·26-s + 1.92·27-s + 1.13·28-s − 0.589·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.011858914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011858914\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 19 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 2 p T + 279 T^{2} - 1996 T^{3} + 279 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - p T + 916 T^{2} - 629 p T^{3} + 916 p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 p T - 747 T^{2} - 100096 T^{3} - 747 p^{3} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 9 T + 1962 T^{2} - 1861 T^{3} + 1962 p^{3} T^{4} - 9 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 84 T + 4659 T^{2} + 4584 T^{3} + 4659 p^{3} T^{4} + 84 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 6309 T^{2} + 155444 T^{3} + 6309 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 92 T + 43023 T^{2} + 5789912 T^{3} + 43023 p^{3} T^{4} + 92 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 109 T + 15884 T^{2} + 3227519 T^{3} + 15884 p^{3} T^{4} - 109 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 245 T + 80954 T^{2} + 26137769 T^{3} + 80954 p^{3} T^{4} + 245 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 688 T + 7203 p T^{2} - 84555424 T^{3} + 7203 p^{4} T^{4} - 688 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 103 T + 159224 T^{2} + 22627771 T^{3} + 159224 p^{3} T^{4} + 103 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 322 T + 145305 T^{2} - 78557908 T^{3} + 145305 p^{3} T^{4} - 322 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 1322 T + 980079 T^{2} - 460609148 T^{3} + 980079 p^{3} T^{4} - 1322 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 252 T + 626829 T^{2} + 103143600 T^{3} + 626829 p^{3} T^{4} + 252 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 435 T + 8826 p T^{2} + 3236059 p T^{3} + 8826 p^{4} T^{4} + 435 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 719 T + 1051816 T^{2} - 6598681 p T^{3} + 1051816 p^{3} T^{4} - 719 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 62 T + 355521 T^{2} + 66663700 T^{3} + 355521 p^{3} T^{4} - 62 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 581 T + 656782 T^{2} + 522695089 T^{3} + 656782 p^{3} T^{4} + 581 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 489 T + 1384836 T^{2} - 430373365 T^{3} + 1384836 p^{3} T^{4} - 489 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2496 T + 3499089 T^{2} - 3226639296 T^{3} + 3499089 p^{3} T^{4} - 2496 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1584 T + 2188023 T^{2} + 1837823112 T^{3} + 2188023 p^{3} T^{4} + 1584 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 974 T + 3028319 T^{2} + 1802087204 T^{3} + 3028319 p^{3} T^{4} + 974 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−7.77333463649440389408113678504, −7.46399773275355779943029623995, −7.42518893633463596843479852862, −7.27385059292175093766341880848, −6.79747114193950797669936125192, −6.38642282094283099263239180693, −6.38194062113948619725866019143, −5.98314385669957276898892914281, −5.59077056247013702326652502629, −5.51272260809394484020133349499, −4.97428247535207312610458086811, −4.68573956356123535059766519944, −4.34693911501265531622379814701, −3.91495861166455511854737824621, −3.56247560226979298258615881366, −3.47032333416412391615602731630, −2.73131682188569530511060743424, −2.60416167487638521292207618060, −2.50910391555321575753976524915, −1.94440808319028050196600936627, −1.81668553488916473749729846344, −1.76835346108614258552362251969, −0.919518570924874569488072148034, −0.791482426939584380428333117250, −0.22930349049629611194478927304,
0.22930349049629611194478927304, 0.791482426939584380428333117250, 0.919518570924874569488072148034, 1.76835346108614258552362251969, 1.81668553488916473749729846344, 1.94440808319028050196600936627, 2.50910391555321575753976524915, 2.60416167487638521292207618060, 2.73131682188569530511060743424, 3.47032333416412391615602731630, 3.56247560226979298258615881366, 3.91495861166455511854737824621, 4.34693911501265531622379814701, 4.68573956356123535059766519944, 4.97428247535207312610458086811, 5.51272260809394484020133349499, 5.59077056247013702326652502629, 5.98314385669957276898892914281, 6.38194062113948619725866019143, 6.38642282094283099263239180693, 6.79747114193950797669936125192, 7.27385059292175093766341880848, 7.42518893633463596843479852862, 7.46399773275355779943029623995, 7.77333463649440389408113678504
