| L(s) = 1 | − 1.78·2-s − 0.436·3-s − 4.81·4-s + 13.8·5-s + 0.779·6-s − 6.26·7-s + 22.8·8-s − 26.8·9-s − 24.6·10-s − 11·11-s + 2.10·12-s + 11.1·14-s − 6.03·15-s − 2.29·16-s + 7.27·17-s + 47.8·18-s − 91.0·19-s − 66.4·20-s + 2.73·21-s + 19.6·22-s + 157.·23-s − 9.98·24-s + 65.6·25-s + 23.4·27-s + 30.1·28-s + 206.·29-s + 10.7·30-s + ⋯ |
| L(s) = 1 | − 0.630·2-s − 0.0840·3-s − 0.601·4-s + 1.23·5-s + 0.0530·6-s − 0.338·7-s + 1.01·8-s − 0.992·9-s − 0.779·10-s − 0.301·11-s + 0.0505·12-s + 0.213·14-s − 0.103·15-s − 0.0358·16-s + 0.103·17-s + 0.626·18-s − 1.09·19-s − 0.743·20-s + 0.0284·21-s + 0.190·22-s + 1.42·23-s − 0.0849·24-s + 0.525·25-s + 0.167·27-s + 0.203·28-s + 1.32·29-s + 0.0654·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.78T + 8T^{2} \) |
| 3 | \( 1 + 0.436T + 27T^{2} \) |
| 5 | \( 1 - 13.8T + 125T^{2} \) |
| 7 | \( 1 + 6.26T + 343T^{2} \) |
| 17 | \( 1 - 7.27T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 8.76T + 2.97e4T^{2} \) |
| 37 | \( 1 - 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 58.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 23.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.21T + 1.03e5T^{2} \) |
| 53 | \( 1 + 554.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 129.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 242.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 398.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 757.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 872.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 773.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 4.22T + 5.71e5T^{2} \) |
| 89 | \( 1 - 290.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 642.T + 9.12e5T^{2} \) |
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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
−8.580045139915145150610777664821, −8.044953799434818625912400933659, −6.80675387302641167091837736505, −6.12133728425004437802695339286, −5.27074120766115271127138512162, −4.58627870484885774988250806423, −3.20639613095207592781727841189, −2.28338859367083926550329549038, −1.11427039837900195448599460096, 0,
1.11427039837900195448599460096, 2.28338859367083926550329549038, 3.20639613095207592781727841189, 4.58627870484885774988250806423, 5.27074120766115271127138512162, 6.12133728425004437802695339286, 6.80675387302641167091837736505, 8.044953799434818625912400933659, 8.580045139915145150610777664821
