| L(s) = 1 | − 18.9·2-s + 230.·4-s − 260.·5-s + 254.·7-s − 1.95e3·8-s + 4.93e3·10-s + 3.60e3·11-s + 1.00e4·13-s − 4.83e3·14-s + 7.38e3·16-s − 4.42e3·17-s + 1.69e4·19-s − 6.01e4·20-s − 6.82e4·22-s − 6.31e4·23-s − 1.02e4·25-s − 1.90e5·26-s + 5.88e4·28-s + 2.74e4·29-s − 2.22e4·31-s + 1.09e5·32-s + 8.37e4·34-s − 6.64e4·35-s + 1.11e5·37-s − 3.21e5·38-s + 5.08e5·40-s − 8.75e4·41-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.80·4-s − 0.932·5-s + 0.280·7-s − 1.34·8-s + 1.56·10-s + 0.815·11-s + 1.27·13-s − 0.470·14-s + 0.451·16-s − 0.218·17-s + 0.567·19-s − 1.68·20-s − 1.36·22-s − 1.08·23-s − 0.130·25-s − 2.13·26-s + 0.506·28-s + 0.208·29-s − 0.133·31-s + 0.591·32-s + 0.365·34-s − 0.261·35-s + 0.362·37-s − 0.950·38-s + 1.25·40-s − 0.198·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 18.9T + 128T^{2} \) |
| 5 | \( 1 + 260.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 254.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.60e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.00e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.42e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.69e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.31e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.74e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.22e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.11e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.75e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.54e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.47e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.10e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.27e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.66e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.63e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.79e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.69e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.64e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.90e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.05e5T + 8.07e13T^{2} \) |
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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
−9.143731540515599564068766999097, −8.350428833771782952774935705436, −7.84211893032804675098574605022, −6.89008877747176988637553132833, −6.00873542289765852334679329433, −4.34899758126947745600879755682, −3.36593475615556424421091939351, −1.85205343190950110219733848289, −1.01118379919317395459933258421, 0,
1.01118379919317395459933258421, 1.85205343190950110219733848289, 3.36593475615556424421091939351, 4.34899758126947745600879755682, 6.00873542289765852334679329433, 6.89008877747176988637553132833, 7.84211893032804675098574605022, 8.350428833771782952774935705436, 9.143731540515599564068766999097
