| L(s) = 1 | + 10.9·2-s + 86.8·4-s − 88.1·5-s + 61.7·7-s + 598.·8-s − 960.·10-s − 423.·11-s − 1.04e3·13-s + 672.·14-s + 3.74e3·16-s + 2.03e3·17-s − 1.53e3·19-s − 7.65e3·20-s − 4.61e3·22-s − 2.98e3·23-s + 4.63e3·25-s − 1.14e4·26-s + 5.36e3·28-s − 115.·29-s − 3.62e3·31-s + 2.16e4·32-s + 2.21e4·34-s − 5.43e3·35-s − 1.50e4·37-s − 1.67e4·38-s − 5.27e4·40-s + 829.·41-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 2.71·4-s − 1.57·5-s + 0.476·7-s + 3.30·8-s − 3.03·10-s − 1.05·11-s − 1.71·13-s + 0.917·14-s + 3.65·16-s + 1.70·17-s − 0.973·19-s − 4.27·20-s − 2.03·22-s − 1.17·23-s + 1.48·25-s − 3.31·26-s + 1.29·28-s − 0.0254·29-s − 0.677·31-s + 3.73·32-s + 3.29·34-s − 0.750·35-s − 1.80·37-s − 1.87·38-s − 5.20·40-s + 0.0770·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 - 10.9T + 32T^{2} \) |
| 5 | \( 1 + 88.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 61.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 423.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.53e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 115.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.50e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 829.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.22e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.61e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.44e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 1.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.81e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.38e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.85e4T + 8.58e9T^{2} \) |
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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
−10.13942463668201896136268871669, −7.954085847262901871219381698552, −7.74201761447994759468359308379, −6.80452026781463420342356302939, −5.39559360084288513179744367594, −4.87373299128814860701329078435, −3.92109110669564594697902738570, −3.11024593864715746992115244528, −1.99566345150654380105243966239, 0,
1.99566345150654380105243966239, 3.11024593864715746992115244528, 3.92109110669564594697902738570, 4.87373299128814860701329078435, 5.39559360084288513179744367594, 6.80452026781463420342356302939, 7.74201761447994759468359308379, 7.954085847262901871219381698552, 10.13942463668201896136268871669
