| L(s) = 1 | − 3·2-s + 3·4-s − 6·5-s − 3·7-s + 18·10-s − 3·11-s − 3·13-s + 9·14-s − 3·16-s − 9·17-s − 3·19-s − 18·20-s + 9·22-s − 6·23-s + 12·25-s + 9·26-s − 9·28-s − 12·29-s − 12·31-s + 6·32-s + 27·34-s + 18·35-s − 3·37-s + 9·38-s + 3·41-s − 12·43-s − 9·44-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s − 2.68·5-s − 1.13·7-s + 5.69·10-s − 0.904·11-s − 0.832·13-s + 2.40·14-s − 3/4·16-s − 2.18·17-s − 0.688·19-s − 4.02·20-s + 1.91·22-s − 1.25·23-s + 12/5·25-s + 1.76·26-s − 1.70·28-s − 2.22·29-s − 2.15·31-s + 1.06·32-s + 4.63·34-s + 3.04·35-s − 0.493·37-s + 1.45·38-s + 0.468·41-s − 1.82·43-s − 1.35·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14348907 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14348907 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 63 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 25 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 63 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 61 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 115 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 225 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 12 T + 114 T^{2} + 639 T^{3} + 114 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 763 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 87 T^{2} + 223 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 3 T + 69 T^{2} - 27 T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1051 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 6 T + 78 T^{2} - 297 T^{3} + 78 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 18 T + 240 T^{2} + 1989 T^{3} + 240 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 21 T + 321 T^{2} - 2799 T^{3} + 321 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 785 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 695 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 51 T^{2} - 279 T^{3} + 51 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 479 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 6 T + 186 T^{2} - 1001 T^{3} + 186 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 222 T^{2} - 945 T^{3} + 222 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 78 T^{2} - 999 T^{3} + 78 p T^{4} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 15 T + 222 T^{2} - 2891 T^{3} + 222 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−11.21094927693477080970868832223, −10.99028265917821185663419975035, −10.69805872945867638656025958998, −10.31564173334467064296540622658, −9.698218293970934472866009874913, −9.563348264986510895218878526500, −9.541383471047957055748250942967, −8.992831124463235558092316781552, −8.651597628554224859418972476930, −8.387826227475013430850662966704, −8.041914134602312911887435922690, −7.77789329088139499425888908988, −7.68569108783688244384418038570, −7.07973079361832528250324311605, −6.91356097673695218347818511037, −6.51730807411975266096373473957, −6.03587855749369941395695055753, −5.15908875112276506092503140885, −5.15289469433362793276361238788, −4.35238690733375582841137514901, −3.91384805902053872584527872735, −3.74567657895574150041096517072, −3.38748189386890677398618911584, −2.27520788694591860789670521352, −2.08831159546667970910620055785, 0, 0, 0,
2.08831159546667970910620055785, 2.27520788694591860789670521352, 3.38748189386890677398618911584, 3.74567657895574150041096517072, 3.91384805902053872584527872735, 4.35238690733375582841137514901, 5.15289469433362793276361238788, 5.15908875112276506092503140885, 6.03587855749369941395695055753, 6.51730807411975266096373473957, 6.91356097673695218347818511037, 7.07973079361832528250324311605, 7.68569108783688244384418038570, 7.77789329088139499425888908988, 8.041914134602312911887435922690, 8.387826227475013430850662966704, 8.651597628554224859418972476930, 8.992831124463235558092316781552, 9.541383471047957055748250942967, 9.563348264986510895218878526500, 9.698218293970934472866009874913, 10.31564173334467064296540622658, 10.69805872945867638656025958998, 10.99028265917821185663419975035, 11.21094927693477080970868832223
