| L(s) = 1 | − 2-s + 9·3-s − 14·4-s − 9·6-s + 16·7-s + 18·8-s + 54·9-s − 33·11-s − 126·12-s + 42·13-s − 16·14-s + 83·16-s + 34·17-s − 54·18-s − 280·19-s + 144·21-s + 33·22-s + 112·23-s + 162·24-s − 42·26-s + 270·27-s − 224·28-s − 290·29-s − 392·31-s − 143·32-s − 297·33-s − 34·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.73·3-s − 7/4·4-s − 0.612·6-s + 0.863·7-s + 0.795·8-s + 2·9-s − 0.904·11-s − 3.03·12-s + 0.896·13-s − 0.305·14-s + 1.29·16-s + 0.485·17-s − 0.707·18-s − 3.38·19-s + 1.49·21-s + 0.319·22-s + 1.01·23-s + 1.37·24-s − 0.316·26-s + 1.92·27-s − 1.51·28-s − 1.85·29-s − 2.27·31-s − 0.789·32-s − 1.56·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + 15 T^{2} + 11 T^{3} + 15 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 16 T + 277 T^{2} - 5168 T^{3} + 277 p^{3} T^{4} - 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 42 T + 1363 T^{2} - 47132 T^{3} + 1363 p^{3} T^{4} - 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 p T + 10079 T^{2} - 155020 T^{3} + 10079 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 280 T + 39201 T^{2} + 3743984 T^{3} + 39201 p^{3} T^{4} + 280 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 112 T + 19477 T^{2} - 809120 T^{3} + 19477 p^{3} T^{4} - 112 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 p T + 46667 T^{2} + 4894124 T^{3} + 46667 p^{3} T^{4} + 10 p^{7} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 392 T + 121533 T^{2} + 23039344 T^{3} + 121533 p^{3} T^{4} + 392 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 570 T + 212899 T^{2} + 56704796 T^{3} + 212899 p^{3} T^{4} + 570 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 662 T + 150855 T^{2} + 22689620 T^{3} + 150855 p^{3} T^{4} + 662 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 68 T + 11145 T^{2} + 9678104 T^{3} + 11145 p^{3} T^{4} - 68 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 264 T + 146893 T^{2} + 68940144 T^{3} + 146893 p^{3} T^{4} + 264 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 94 T + 388819 T^{2} - 25584884 T^{3} + 388819 p^{3} T^{4} - 94 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 612 T + 191353 T^{2} + 89255576 T^{3} + 191353 p^{3} T^{4} + 612 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 582 T + 700987 T^{2} + 242850884 T^{3} + 700987 p^{3} T^{4} + 582 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 940 T + 512801 T^{2} + 170936200 T^{3} + 512801 p^{3} T^{4} + 940 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 1616 T + 1744933 T^{2} + 1196967136 T^{3} + 1744933 p^{3} T^{4} + 1616 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 738 T + 1335919 T^{2} + 586235356 T^{3} + 1335919 p^{3} T^{4} + 738 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 124 T + 1293245 T^{2} - 95402344 T^{3} + 1293245 p^{3} T^{4} - 124 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1232 T + 2008521 T^{2} - 1393226768 T^{3} + 2008521 p^{3} T^{4} - 1232 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 838 T + 1110375 T^{2} - 349581812 T^{3} + 1110375 p^{3} T^{4} - 838 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 90 T + 441679 T^{2} + 759889556 T^{3} + 441679 p^{3} T^{4} - 90 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.975257984627867981597137714799, −8.756886792663463848002055456392, −8.664874130697926529337598270123, −8.644008272762849038412990189364, −8.062074708593717934400554198884, −7.83612037750024537156086712475, −7.55847616218115568494835181138, −7.44909690622839603019137495634, −6.87360478448751635380286870537, −6.72604814050844483706955404452, −6.04175695660354099526665503266, −6.01966312603529477032849909912, −5.32961516232190930103421756989, −5.10611389301653612594103417886, −4.76564008235122752127719055300, −4.68472080083570909399073377723, −4.07271715208151433876052641457, −3.85092018567788053733613541436, −3.69672774747570427612935804381, −3.08737524024092746834732223996, −3.06091685741671221142367814818, −2.12551233059698608939943653504, −1.90538860750957338173778733004, −1.61551328094500872726920373329, −1.34100164728806702134706110119, 0, 0, 0,
1.34100164728806702134706110119, 1.61551328094500872726920373329, 1.90538860750957338173778733004, 2.12551233059698608939943653504, 3.06091685741671221142367814818, 3.08737524024092746834732223996, 3.69672774747570427612935804381, 3.85092018567788053733613541436, 4.07271715208151433876052641457, 4.68472080083570909399073377723, 4.76564008235122752127719055300, 5.10611389301653612594103417886, 5.32961516232190930103421756989, 6.01966312603529477032849909912, 6.04175695660354099526665503266, 6.72604814050844483706955404452, 6.87360478448751635380286870537, 7.44909690622839603019137495634, 7.55847616218115568494835181138, 7.83612037750024537156086712475, 8.062074708593717934400554198884, 8.644008272762849038412990189364, 8.664874130697926529337598270123, 8.756886792663463848002055456392, 8.975257984627867981597137714799
