| L(s) = 1 | + 6·2-s − 3·3-s + 24·4-s − 15·5-s − 18·6-s + 3·7-s + 80·8-s − 48·9-s − 90·10-s − 63·11-s − 72·12-s − 54·13-s + 18·14-s + 45·15-s + 240·16-s − 36·17-s − 288·18-s − 90·19-s − 360·20-s − 9·21-s − 378·22-s − 282·23-s − 240·24-s + 150·25-s − 324·26-s + 143·27-s + 72·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.577·3-s + 3·4-s − 1.34·5-s − 1.22·6-s + 0.161·7-s + 3.53·8-s − 1.77·9-s − 2.84·10-s − 1.72·11-s − 1.73·12-s − 1.15·13-s + 0.343·14-s + 0.774·15-s + 15/4·16-s − 0.513·17-s − 3.77·18-s − 1.08·19-s − 4.02·20-s − 0.0935·21-s − 3.66·22-s − 2.55·23-s − 2.04·24-s + 6/5·25-s − 2.44·26-s + 1.01·27-s + 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50653000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50653000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 37 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T + 19 p T^{2} + 172 T^{3} + 19 p^{4} T^{4} + p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 645 T^{2} + 1004 T^{3} + 645 p^{3} T^{4} - 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 63 T + 4173 T^{2} + 144786 T^{3} + 4173 p^{3} T^{4} + 63 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 54 T + 3711 T^{2} + 19356 p T^{3} + 3711 p^{3} T^{4} + 54 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 36 T + 12219 T^{2} + 381576 T^{3} + 12219 p^{3} T^{4} + 36 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 90 T + 23187 T^{2} + 1259220 T^{3} + 23187 p^{3} T^{4} + 90 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 282 T + 2403 p T^{2} + 7072012 T^{3} + 2403 p^{4} T^{4} + 282 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 234 T + 30507 T^{2} + 1611324 T^{3} + 30507 p^{3} T^{4} + 234 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 90 T + 77295 T^{2} - 5523108 T^{3} + 77295 p^{3} T^{4} - 90 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 117 T + 175011 T^{2} + 16663998 T^{3} + 175011 p^{3} T^{4} + 117 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 570 T + 128553 T^{2} + 26579932 T^{3} + 128553 p^{3} T^{4} + 570 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 573 T + 319005 T^{2} + 102029276 T^{3} + 319005 p^{3} T^{4} + 573 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 231 T + 384183 T^{2} + 68405762 T^{3} + 384183 p^{3} T^{4} + 231 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 54 T + 610899 T^{2} + 22174284 T^{3} + 610899 p^{3} T^{4} + 54 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 576 T + 733107 T^{2} + 252303792 T^{3} + 733107 p^{3} T^{4} + 576 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 774 T + 825699 T^{2} - 463956036 T^{3} + 825699 p^{3} T^{4} - 774 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 219 T + 1016037 T^{2} - 150472474 T^{3} + 1016037 p^{3} T^{4} - 219 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 675 T + 1170939 T^{2} - 523874034 T^{3} + 1170939 p^{3} T^{4} - 675 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 150 T + 1002399 T^{2} - 4813828 T^{3} + 1002399 p^{3} T^{4} + 150 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 219 T + 367461 T^{2} - 153632684 T^{3} + 367461 p^{3} T^{4} + 219 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 108 T + 1585023 T^{2} + 273140664 T^{3} + 1585023 p^{3} T^{4} + 108 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2004 T + 3521031 T^{2} - 3560605736 T^{3} + 3521031 p^{3} T^{4} - 2004 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−10.68359942339214464099875865614, −9.964796712117066866886723371230, −9.793028726593946794739025627668, −9.542082075898300645453425376152, −8.586563497042660455951730681886, −8.480937243312517424535706383650, −8.215304908107118964507888255235, −7.85683254799320125588166004776, −7.68369121567299892189707166147, −7.41181570315475921276827664098, −6.63670859338898073263649220168, −6.53441350185508405038213263676, −6.36728132351204232941560772685, −5.69110224098654436215518463819, −5.59868749274801597609871403058, −5.13053035857621825639173471620, −4.88723220934143153629789185619, −4.64106223543035108213198899086, −4.23298796427742447239679184010, −3.64550428498625711152891528483, −3.29062093287533573712937809265, −3.17482101005571655566802977134, −2.36009267612318879770861363290, −2.23837523350959499443397484733, −1.72752765894579084617212723474, 0, 0, 0,
1.72752765894579084617212723474, 2.23837523350959499443397484733, 2.36009267612318879770861363290, 3.17482101005571655566802977134, 3.29062093287533573712937809265, 3.64550428498625711152891528483, 4.23298796427742447239679184010, 4.64106223543035108213198899086, 4.88723220934143153629789185619, 5.13053035857621825639173471620, 5.59868749274801597609871403058, 5.69110224098654436215518463819, 6.36728132351204232941560772685, 6.53441350185508405038213263676, 6.63670859338898073263649220168, 7.41181570315475921276827664098, 7.68369121567299892189707166147, 7.85683254799320125588166004776, 8.215304908107118964507888255235, 8.480937243312517424535706383650, 8.586563497042660455951730681886, 9.542082075898300645453425376152, 9.793028726593946794739025627668, 9.964796712117066866886723371230, 10.68359942339214464099875865614
