| L(s) = 1 | + 2·3-s − 4·9-s − 11-s − 5·13-s − 2·17-s + 3·19-s + 8·23-s − 13·27-s − 7·29-s + 5·31-s − 2·33-s − 5·37-s − 10·39-s + 41-s + 5·43-s + 3·47-s − 14·49-s − 4·51-s − 19·53-s + 6·57-s − 10·59-s − 17·61-s − 67-s + 16·69-s + 19·71-s + 73-s − 18·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4/3·9-s − 0.301·11-s − 1.38·13-s − 0.485·17-s + 0.688·19-s + 1.66·23-s − 2.50·27-s − 1.29·29-s + 0.898·31-s − 0.348·33-s − 0.821·37-s − 1.60·39-s + 0.156·41-s + 0.762·43-s + 0.437·47-s − 2·49-s − 0.560·51-s − 2.60·53-s + 0.794·57-s − 1.30·59-s − 2.17·61-s − 0.122·67-s + 1.92·69-s + 2.25·71-s + 0.117·73-s − 2.02·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $A_4\times C_2$ | \( 1 - 2 T + 8 T^{2} - 11 T^{3} + 8 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 2 p T^{2} + p T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + T + 17 T^{2} + 35 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 5 T + 45 T^{2} + 131 T^{3} + 45 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 2 T + 36 T^{2} + 81 T^{3} + 36 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 8 T + 88 T^{2} - 381 T^{3} + 88 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 7 T + 45 T^{2} + 315 T^{3} + 45 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 5 T + 57 T^{2} - 353 T^{3} + 57 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 5 T + 103 T^{2} + 371 T^{3} + 103 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - T + 9 T^{2} + 339 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 5 T + 72 T^{2} - 137 T^{3} + 72 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 137 T^{2} - 269 T^{3} + 137 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 19 T + 214 T^{2} + 1707 T^{3} + 214 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 10 T + 145 T^{2} + 852 T^{3} + 145 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 17 T + 249 T^{2} + 2033 T^{3} + 249 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + T + 59 T^{2} + 693 T^{3} + 59 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 19 T + 268 T^{2} - 2391 T^{3} + 268 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - T + 49 T^{2} + 317 T^{3} + 49 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 18 T + 324 T^{2} + 2941 T^{3} + 324 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 13 T + 247 T^{2} - 2019 T^{3} + 247 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 2 T + 168 T^{2} - 75 T^{3} + 168 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 5 T + 297 T^{2} + 971 T^{3} + 297 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.53263564810710381778577625302, −7.21069828517382856031726027960, −6.81297594328714632824049897425, −6.71314180768639174777900977224, −6.22975453060353168147993980750, −6.13520921682050284307439250236, −6.12799676800734252005033792467, −5.61556599079074111822971026606, −5.27478353678068160627102618998, −5.19560533850131874964330116754, −4.87055392071616813395011469202, −4.83417403422460361110828508661, −4.59833976195292091412427318397, −4.07579865150227238079918520742, −3.83997453568447539388446067130, −3.53881088741684522168346493504, −3.24245076796286164771341758708, −3.01456189507633647254229079886, −2.94635701566468527609621009932, −2.60396422175602822916396223624, −2.28629400595512651982593412862, −2.24242504134781421832748328437, −1.66004940948084921898970076954, −1.19795095115109218900533763898, −1.19158975922590861684963161738, 0, 0, 0,
1.19158975922590861684963161738, 1.19795095115109218900533763898, 1.66004940948084921898970076954, 2.24242504134781421832748328437, 2.28629400595512651982593412862, 2.60396422175602822916396223624, 2.94635701566468527609621009932, 3.01456189507633647254229079886, 3.24245076796286164771341758708, 3.53881088741684522168346493504, 3.83997453568447539388446067130, 4.07579865150227238079918520742, 4.59833976195292091412427318397, 4.83417403422460361110828508661, 4.87055392071616813395011469202, 5.19560533850131874964330116754, 5.27478353678068160627102618998, 5.61556599079074111822971026606, 6.12799676800734252005033792467, 6.13520921682050284307439250236, 6.22975453060353168147993980750, 6.71314180768639174777900977224, 6.81297594328714632824049897425, 7.21069828517382856031726027960, 7.53263564810710381778577625302
