| 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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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
−7.87839943554406021193752701519, −7.65817198439213465915424773738, −7.44905230758568324751638489480, −7.20783576111380342442012195739, −6.54928791323552567030951085607, −6.54762569082630123050346536011, −6.47134108718901015267226278878, −6.11535773269686048163206627505, −6.02844590069196671982796596644, −5.58988454803544394526593744880, −5.24178893722597618389588252241, −5.15981203909107988450380637301, −4.92060896877526683111757824735, −4.74116751891594191722940124403, −4.46731649630910423975293776314, −4.14985879806247405964625764716, −3.76419964941232294033542249397, −3.37147577894433230143714113089, −3.22213868692223366611329003916, −2.67005346838084301496189808136, −2.31377187247067848000562867497, −2.16132234553381211054735264234, −1.56419415318510693927094167028, −1.53016274877462214274435150437, −1.14651442190544055386270054683, 0, 0, 0,
1.14651442190544055386270054683, 1.53016274877462214274435150437, 1.56419415318510693927094167028, 2.16132234553381211054735264234, 2.31377187247067848000562867497, 2.67005346838084301496189808136, 3.22213868692223366611329003916, 3.37147577894433230143714113089, 3.76419964941232294033542249397, 4.14985879806247405964625764716, 4.46731649630910423975293776314, 4.74116751891594191722940124403, 4.92060896877526683111757824735, 5.15981203909107988450380637301, 5.24178893722597618389588252241, 5.58988454803544394526593744880, 6.02844590069196671982796596644, 6.11535773269686048163206627505, 6.47134108718901015267226278878, 6.54762569082630123050346536011, 6.54928791323552567030951085607, 7.20783576111380342442012195739, 7.44905230758568324751638489480, 7.65817198439213465915424773738, 7.87839943554406021193752701519
