| L(s) = 1 | − 5·2-s + 9·4-s − 15·5-s − 4·7-s − 15·8-s + 75·10-s − 5·11-s + 7·13-s + 20·14-s + 53·16-s − 155·17-s − 50·19-s − 135·20-s + 25·22-s − 285·23-s + 150·25-s − 35·26-s − 36·28-s − 115·29-s − 115·31-s − 155·32-s + 775·34-s + 60·35-s − 384·37-s + 250·38-s + 225·40-s − 580·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 9/8·4-s − 1.34·5-s − 0.215·7-s − 0.662·8-s + 2.37·10-s − 0.137·11-s + 0.149·13-s + 0.381·14-s + 0.828·16-s − 2.21·17-s − 0.603·19-s − 1.50·20-s + 0.242·22-s − 2.58·23-s + 6/5·25-s − 0.264·26-s − 0.242·28-s − 0.736·29-s − 0.666·31-s − 0.856·32-s + 3.90·34-s + 0.289·35-s − 1.70·37-s + 1.06·38-s + 0.889·40-s − 2.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2460375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2460375 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 5 T + p^{4} T^{2} + 25 p T^{3} + p^{7} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 234 T^{2} - 5554 T^{3} + 234 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5 T + 1105 T^{2} - 17950 T^{3} + 1105 p^{3} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 7 T + 5822 T^{2} - 37183 T^{3} + 5822 p^{3} T^{4} - 7 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 155 T + 20347 T^{2} + 1564790 T^{3} + 20347 p^{3} T^{4} + 155 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 50 T + 206 p T^{2} + 317888 T^{3} + 206 p^{4} T^{4} + 50 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 285 T + 52701 T^{2} + 6381690 T^{3} + 52701 p^{3} T^{4} + 285 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 115 T + 25759 T^{2} - 830870 T^{3} + 25759 p^{3} T^{4} + 115 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 115 T + 60141 T^{2} + 5913626 T^{3} + 60141 p^{3} T^{4} + 115 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 384 T + 84036 T^{2} + 16234306 T^{3} + 84036 p^{3} T^{4} + 384 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 580 T + 296575 T^{2} + 83865640 T^{3} + 296575 p^{3} T^{4} + 580 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 797 T + 381041 T^{2} + 121376222 T^{3} + 381041 p^{3} T^{4} + 797 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 145 T + 62317 T^{2} - 44496910 T^{3} + 62317 p^{3} T^{4} - 145 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 400 T + 388459 T^{2} - 106443280 T^{3} + 388459 p^{3} T^{4} - 400 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 380 T + 588745 T^{2} + 150882920 T^{3} + 588745 p^{3} T^{4} + 380 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 152 T + 593468 T^{2} + 63933158 T^{3} + 593468 p^{3} T^{4} + 152 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 658290 T^{2} + 19566248 T^{3} + 658290 p^{3} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 40 T + 396361 T^{2} - 187438400 T^{3} + 396361 p^{3} T^{4} + 40 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 980 T + 1431584 T^{2} + 778921274 T^{3} + 1431584 p^{3} T^{4} + 980 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1013 T + 1531332 T^{2} - 908300089 T^{3} + 1531332 p^{3} T^{4} - 1013 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 270 T + 1413933 T^{2} + 224225820 T^{3} + 1413933 p^{3} T^{4} + 270 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1020 T + 2161707 T^{2} + 1313072760 T^{3} + 2161707 p^{3} T^{4} + 1020 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 720 T + 2703456 T^{2} - 1286818562 T^{3} + 2703456 p^{3} T^{4} - 720 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−11.72153793629543734917912107376, −11.67814637656873645324046210556, −11.04168591207642088134434207604, −10.91472828531969028664698990237, −10.34141985996437328778945871911, −10.06327599451344043598990676847, −9.977270890274071246218878528327, −9.259689462649478747742999760843, −8.948622735300731405871099107539, −8.764726657610715107773695003814, −8.393860479009441564492549028495, −8.182743296263062209903294619836, −7.84159135132084991879744901349, −7.29905575078858358403604386712, −6.89033249329496026564965502695, −6.47360098252878396449022845602, −6.37366784945241391825937438286, −5.39309285183651874910948327425, −5.18630306595890004521382144446, −4.36119619187413278627252558782, −4.01605913178410730470129241910, −3.59870478363771943305380763226, −2.99192579651549168603320142315, −1.97962745682211263790175915637, −1.71328184353382354875338618041, 0, 0, 0,
1.71328184353382354875338618041, 1.97962745682211263790175915637, 2.99192579651549168603320142315, 3.59870478363771943305380763226, 4.01605913178410730470129241910, 4.36119619187413278627252558782, 5.18630306595890004521382144446, 5.39309285183651874910948327425, 6.37366784945241391825937438286, 6.47360098252878396449022845602, 6.89033249329496026564965502695, 7.29905575078858358403604386712, 7.84159135132084991879744901349, 8.182743296263062209903294619836, 8.393860479009441564492549028495, 8.764726657610715107773695003814, 8.948622735300731405871099107539, 9.259689462649478747742999760843, 9.977270890274071246218878528327, 10.06327599451344043598990676847, 10.34141985996437328778945871911, 10.91472828531969028664698990237, 11.04168591207642088134434207604, 11.67814637656873645324046210556, 11.72153793629543734917912107376
