| L(s) = 1 | − 1.32·2-s + 3.35·3-s − 6.25·4-s + 2.79·5-s − 4.43·6-s + 18.8·8-s − 15.7·9-s − 3.68·10-s − 29.3·11-s − 20.9·12-s + 76.2·13-s + 9.36·15-s + 25.2·16-s − 47.1·17-s + 20.7·18-s + 149.·19-s − 17.4·20-s + 38.7·22-s − 36.2·23-s + 63.1·24-s − 117.·25-s − 100.·26-s − 143.·27-s + 21.9·29-s − 12.3·30-s − 248.·31-s − 183.·32-s + ⋯ |
| L(s) = 1 | − 0.466·2-s + 0.645·3-s − 0.782·4-s + 0.249·5-s − 0.301·6-s + 0.831·8-s − 0.582·9-s − 0.116·10-s − 0.804·11-s − 0.505·12-s + 1.62·13-s + 0.161·15-s + 0.393·16-s − 0.673·17-s + 0.272·18-s + 1.81·19-s − 0.195·20-s + 0.375·22-s − 0.328·23-s + 0.537·24-s − 0.937·25-s − 0.758·26-s − 1.02·27-s + 0.140·29-s − 0.0752·30-s − 1.43·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 47 | \( 1 + 47T \) |
| good | 2 | \( 1 + 1.32T + 8T^{2} \) |
| 3 | \( 1 - 3.35T + 27T^{2} \) |
| 5 | \( 1 - 2.79T + 125T^{2} \) |
| 11 | \( 1 + 29.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 149.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 36.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 21.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 248.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 231.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 426.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 14.9T + 7.95e4T^{2} \) |
| 53 | \( 1 - 190.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 884.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 650.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 636.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 912.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 676.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 501.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 979.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 643.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3T + 9.12e5T^{2} \) |
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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
−8.317633239154504848049378095444, −7.902013314312108568640627689212, −6.91399050193159409995824152435, −5.64062225793781579151867497050, −5.33352052700302158996842848365, −3.95812999661558750370933204072, −3.41547236318350055530005912029, −2.24685250049353438277744240770, −1.15822906632934587237455381954, 0,
1.15822906632934587237455381954, 2.24685250049353438277744240770, 3.41547236318350055530005912029, 3.95812999661558750370933204072, 5.33352052700302158996842848365, 5.64062225793781579151867497050, 6.91399050193159409995824152435, 7.902013314312108568640627689212, 8.317633239154504848049378095444
