| L(s) = 1 | − 4.86·2-s − 8.29·4-s − 66.7·5-s − 185.·7-s + 196.·8-s + 324.·10-s + 563.·11-s + 767.·13-s + 902.·14-s − 689.·16-s + 1.73e3·17-s − 2.28e3·19-s + 553.·20-s − 2.74e3·22-s + 529·23-s + 1.32e3·25-s − 3.73e3·26-s + 1.53e3·28-s − 3.57e3·29-s + 2.83e3·31-s − 2.92e3·32-s − 8.45e3·34-s + 1.23e4·35-s + 4.49e3·37-s + 1.11e4·38-s − 1.30e4·40-s − 1.62e4·41-s + ⋯ |
| L(s) = 1 | − 0.860·2-s − 0.259·4-s − 1.19·5-s − 1.42·7-s + 1.08·8-s + 1.02·10-s + 1.40·11-s + 1.25·13-s + 1.23·14-s − 0.673·16-s + 1.45·17-s − 1.45·19-s + 0.309·20-s − 1.20·22-s + 0.208·23-s + 0.425·25-s − 1.08·26-s + 0.370·28-s − 0.789·29-s + 0.528·31-s − 0.504·32-s − 1.25·34-s + 1.70·35-s + 0.539·37-s + 1.24·38-s − 1.29·40-s − 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 4.86T + 32T^{2} \) |
| 5 | \( 1 + 66.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 185.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 563.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 767.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.73e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.28e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 3.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.49e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.62e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.05e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.33e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.10e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.86e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.75e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.71e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.86987636030994900465278216803, −9.876079787501189027778791938527, −8.986287923855840259070250123783, −8.243826110684528029502789322781, −7.11036084080654841433229083267, −6.09108389845065231021157849507, −4.11932344572590902755407838216, −3.52320849572116044224915082220, −1.14484825701225491126856214473, 0,
1.14484825701225491126856214473, 3.52320849572116044224915082220, 4.11932344572590902755407838216, 6.09108389845065231021157849507, 7.11036084080654841433229083267, 8.243826110684528029502789322781, 8.986287923855840259070250123783, 9.876079787501189027778791938527, 10.86987636030994900465278216803
