| L(s) = 1 | − 3·2-s + 3·4-s + 3·5-s − 9·10-s − 6·11-s + 3·13-s − 3·16-s + 6·17-s + 3·19-s + 9·20-s + 18·22-s − 12·23-s − 6·25-s − 9·26-s − 9·29-s + 3·31-s + 6·32-s − 18·34-s − 3·37-s − 9·38-s − 3·43-s − 18·44-s + 36·46-s + 3·47-s + 18·50-s + 9·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s + 1.34·5-s − 2.84·10-s − 1.80·11-s + 0.832·13-s − 3/4·16-s + 1.45·17-s + 0.688·19-s + 2.01·20-s + 3.83·22-s − 2.50·23-s − 6/5·25-s − 1.76·26-s − 1.67·29-s + 0.538·31-s + 1.06·32-s − 3.08·34-s − 0.493·37-s − 1.45·38-s − 0.457·43-s − 2.71·44-s + 5.30·46-s + 0.437·47-s + 2.54·50-s + 1.24·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 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 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 27 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 135 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 6 T^{2} + 29 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 207 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 97 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 12 T + 96 T^{2} + 549 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 9 T + 51 T^{2} + 189 T^{3} + 51 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} + 137 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} - 101 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 114 T^{2} - 9 T^{3} + 114 p T^{4} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 3 T + 123 T^{2} + 259 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 87 T^{2} - 333 T^{3} + 87 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 639 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 405 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 6 T + 168 T^{2} + 713 T^{3} + 168 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 12 T + 222 T^{2} + 1591 T^{3} + 222 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 159 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 21 T + 303 T^{2} - 2797 T^{3} + 303 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 21 T + 357 T^{2} + 3499 T^{3} + 357 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 18 T + 294 T^{2} + 2997 T^{3} + 294 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 12 T + 204 T^{2} - 1323 T^{3} + 204 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 259 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.992090855002788250884143295333, −7.88257564179225785873658819407, −7.51094368633229295978744705978, −7.49220041548260251899463016034, −6.89233901477527284178078057318, −6.77955314107666130717874850583, −6.25797885895122915719834617233, −6.13690244593135736926558396757, −5.85625023457793339307969597991, −5.75530687322630799781456201934, −5.33830569570120607014691745028, −5.22402256795397575435801949308, −5.21945632600111537781203017980, −4.45200080682492023537994639691, −4.13285591402968959231628286641, −4.08598808125790528385456551520, −3.67717395067779387372885871054, −3.18923650839235621126449953906, −3.08581482555122926593331646215, −2.60630761347350041461782892244, −2.16553015533315645081151428266, −2.15210019887370703910337236471, −1.53046127651340275301785399492, −1.37115831227581009690717062612, −1.12977708143744341949973865999, 0, 0, 0,
1.12977708143744341949973865999, 1.37115831227581009690717062612, 1.53046127651340275301785399492, 2.15210019887370703910337236471, 2.16553015533315645081151428266, 2.60630761347350041461782892244, 3.08581482555122926593331646215, 3.18923650839235621126449953906, 3.67717395067779387372885871054, 4.08598808125790528385456551520, 4.13285591402968959231628286641, 4.45200080682492023537994639691, 5.21945632600111537781203017980, 5.22402256795397575435801949308, 5.33830569570120607014691745028, 5.75530687322630799781456201934, 5.85625023457793339307969597991, 6.13690244593135736926558396757, 6.25797885895122915719834617233, 6.77955314107666130717874850583, 6.89233901477527284178078057318, 7.49220041548260251899463016034, 7.51094368633229295978744705978, 7.88257564179225785873658819407, 7.992090855002788250884143295333
