| L(s) = 1 | − 1.26·2-s + 0.467·3-s − 0.410·4-s − 2.13·5-s − 0.589·6-s + 2.81·7-s + 3.03·8-s − 2.78·9-s + 2.69·10-s + 11-s − 0.191·12-s − 2.46·13-s − 3.54·14-s − 15-s − 3.01·16-s + 1.13·17-s + 3.50·18-s − 0.0997·19-s + 0.878·20-s + 1.31·21-s − 1.26·22-s − 0.527·23-s + 1.42·24-s − 0.425·25-s + 3.11·26-s − 2.70·27-s − 1.15·28-s + ⋯ |
| L(s) = 1 | − 0.891·2-s + 0.269·3-s − 0.205·4-s − 0.956·5-s − 0.240·6-s + 1.06·7-s + 1.07·8-s − 0.927·9-s + 0.852·10-s + 0.301·11-s − 0.0554·12-s − 0.684·13-s − 0.946·14-s − 0.258·15-s − 0.752·16-s + 0.276·17-s + 0.826·18-s − 0.0228·19-s + 0.196·20-s + 0.286·21-s − 0.268·22-s − 0.109·23-s + 0.290·24-s − 0.0850·25-s + 0.610·26-s − 0.520·27-s − 0.218·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.26T + 2T^{2} \) |
| 3 | \( 1 - 0.467T + 3T^{2} \) |
| 5 | \( 1 + 2.13T + 5T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 + 0.0997T + 19T^{2} \) |
| 23 | \( 1 + 0.527T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 6.71T + 37T^{2} \) |
| 41 | \( 1 + 6.85T + 41T^{2} \) |
| 43 | \( 1 + 8.54T + 43T^{2} \) |
| 47 | \( 1 + 0.149T + 47T^{2} \) |
| 53 | \( 1 + 8.21T + 53T^{2} \) |
| 59 | \( 1 - 0.568T + 59T^{2} \) |
| 67 | \( 1 + 2.87T + 67T^{2} \) |
| 71 | \( 1 + 0.962T + 71T^{2} \) |
| 73 | \( 1 + 2.78T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 1.78T + 83T^{2} \) |
| 89 | \( 1 - 2.04T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−9.993099995948213204588888129093, −8.963447619876776414894575336859, −8.328442378475147257934650496662, −7.85209023236856059313405375187, −6.95027774940935982446546029834, −5.32004334725503082438528115541, −4.52454673920528179282044836093, −3.36477418145787138409837655988, −1.72764744079302892728895887532, 0,
1.72764744079302892728895887532, 3.36477418145787138409837655988, 4.52454673920528179282044836093, 5.32004334725503082438528115541, 6.95027774940935982446546029834, 7.85209023236856059313405375187, 8.328442378475147257934650496662, 8.963447619876776414894575336859, 9.993099995948213204588888129093
