| L(s) = 1 | + 0.803·2-s + 9·3-s − 31.3·4-s + 37.1·5-s + 7.23·6-s − 118.·7-s − 50.9·8-s + 81·9-s + 29.8·10-s + 452.·11-s − 282.·12-s − 1.07e3·13-s − 95.2·14-s + 334.·15-s + 962.·16-s + 65.1·18-s + 1.54e3·19-s − 1.16e3·20-s − 1.06e3·21-s + 364.·22-s + 1.28e3·23-s − 458.·24-s − 1.74e3·25-s − 864.·26-s + 729·27-s + 3.71e3·28-s + 1.52e3·29-s + ⋯ |
| L(s) = 1 | + 0.142·2-s + 0.577·3-s − 0.979·4-s + 0.665·5-s + 0.0820·6-s − 0.914·7-s − 0.281·8-s + 0.333·9-s + 0.0945·10-s + 1.12·11-s − 0.565·12-s − 1.76·13-s − 0.129·14-s + 0.384·15-s + 0.939·16-s + 0.0473·18-s + 0.982·19-s − 0.651·20-s − 0.527·21-s + 0.160·22-s + 0.506·23-s − 0.162·24-s − 0.557·25-s − 0.250·26-s + 0.192·27-s + 0.895·28-s + 0.337·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.803T + 32T^{2} \) |
| 5 | \( 1 - 37.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 118.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 452.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.07e3T + 3.71e5T^{2} \) |
| 19 | \( 1 - 1.54e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.28e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.52e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.82e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.21e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.84e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.94e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.23e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.31e5T + 8.58e9T^{2} \) |
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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
−9.335801768465084722825521149181, −8.298241067428596778703046086292, −7.26490206110617352863327903251, −6.41013699655380244189042498076, −5.35828462878161646886772229201, −4.49393422885210918376018828545, −3.47270644522953153809271858623, −2.61374520456940982515683962935, −1.24651114034337198562428089069, 0,
1.24651114034337198562428089069, 2.61374520456940982515683962935, 3.47270644522953153809271858623, 4.49393422885210918376018828545, 5.35828462878161646886772229201, 6.41013699655380244189042498076, 7.26490206110617352863327903251, 8.298241067428596778703046086292, 9.335801768465084722825521149181
