| L(s) = 1 | − 2.30·3-s − 2·4-s − 1.30·5-s + 2.30·9-s − 4.30·11-s + 4.60·12-s + 2.30·13-s + 3·15-s + 4·16-s + 3.90·17-s + 3.60·19-s + 2.60·20-s − 5.60·23-s − 3.30·25-s + 1.60·27-s + 5.60·29-s + 2.30·31-s + 9.90·33-s − 4.60·36-s + 7.21·37-s − 5.30·39-s + 41-s − 9.60·43-s + 8.60·44-s − 3.00·45-s + 9·47-s − 9.21·48-s + ⋯ |
| L(s) = 1 | − 1.32·3-s − 4-s − 0.582·5-s + 0.767·9-s − 1.29·11-s + 1.32·12-s + 0.638·13-s + 0.774·15-s + 16-s + 0.947·17-s + 0.827·19-s + 0.582·20-s − 1.16·23-s − 0.660·25-s + 0.308·27-s + 1.04·29-s + 0.413·31-s + 1.72·33-s − 0.767·36-s + 1.18·37-s − 0.849·39-s + 0.156·41-s − 1.46·43-s + 1.29·44-s − 0.447·45-s + 1.31·47-s − 1.32·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 11 | \( 1 + 4.30T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 3.90T + 17T^{2} \) |
| 19 | \( 1 - 3.60T + 19T^{2} \) |
| 23 | \( 1 + 5.60T + 23T^{2} \) |
| 29 | \( 1 - 5.60T + 29T^{2} \) |
| 31 | \( 1 - 2.30T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 43 | \( 1 + 9.60T + 43T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 - 9.90T + 53T^{2} \) |
| 59 | \( 1 + 0.394T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 9.30T + 73T^{2} \) |
| 79 | \( 1 + 17.4T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 0.394T + 89T^{2} \) |
| 97 | \( 1 + 9.30T + 97T^{2} \) |
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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
−8.584337313748584171008274975841, −8.011964882236786287447004917189, −7.30630530579428335913130207460, −5.96693530228955912877585919749, −5.64135080796154896978556221998, −4.79040531856481382664727393853, −4.04935865377997381264276759061, −2.97327408335083045607922851207, −1.04438508788416614721215932322, 0,
1.04438508788416614721215932322, 2.97327408335083045607922851207, 4.04935865377997381264276759061, 4.79040531856481382664727393853, 5.64135080796154896978556221998, 5.96693530228955912877585919749, 7.30630530579428335913130207460, 8.011964882236786287447004917189, 8.584337313748584171008274975841
