| L(s) = 1 | + 3.59·2-s − 3·3-s + 4.89·4-s − 5·5-s − 10.7·6-s − 16.1·7-s − 11.1·8-s + 9·9-s − 17.9·10-s + 11·11-s − 14.6·12-s − 54.1·13-s − 57.9·14-s + 15·15-s − 79.2·16-s − 107.·17-s + 32.3·18-s + 48.7·19-s − 24.4·20-s + 48.4·21-s + 39.4·22-s + 11.9·23-s + 33.4·24-s + 25·25-s − 194.·26-s − 27·27-s − 78.9·28-s + ⋯ |
| L(s) = 1 | + 1.26·2-s − 0.577·3-s + 0.611·4-s − 0.447·5-s − 0.732·6-s − 0.871·7-s − 0.493·8-s + 0.333·9-s − 0.567·10-s + 0.301·11-s − 0.353·12-s − 1.15·13-s − 1.10·14-s + 0.258·15-s − 1.23·16-s − 1.52·17-s + 0.423·18-s + 0.588·19-s − 0.273·20-s + 0.503·21-s + 0.382·22-s + 0.108·23-s + 0.284·24-s + 0.200·25-s − 1.46·26-s − 0.192·27-s − 0.533·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 3.59T + 8T^{2} \) |
| 7 | \( 1 + 16.1T + 343T^{2} \) |
| 13 | \( 1 + 54.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 11.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 239.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 82.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 21.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 124.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 224.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 186.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 233.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 232.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 163.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 876.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 733.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 588.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.54e3T + 9.12e5T^{2} \) |
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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
−12.11325842279889617853715227935, −11.31503634546292837979045754117, −9.987695812572941867857313303660, −8.883988044396467932812105532638, −7.11217076462862682062123649881, −6.33632740601128452448267112696, −5.06730892068729817480269668819, −4.15210514423586779419605478593, −2.79927966384044821933287276777, 0,
2.79927966384044821933287276777, 4.15210514423586779419605478593, 5.06730892068729817480269668819, 6.33632740601128452448267112696, 7.11217076462862682062123649881, 8.883988044396467932812105532638, 9.987695812572941867857313303660, 11.31503634546292837979045754117, 12.11325842279889617853715227935
