| L(s) = 1 | − 2.77·3-s + 5·5-s + 4.22·7-s − 19.3·9-s + 33.7·11-s − 42.9·13-s − 13.8·15-s + 7.49·17-s + 25.7·19-s − 11.7·21-s + 23·23-s + 25·25-s + 128.·27-s − 47.4·29-s − 65.0·31-s − 93.4·33-s + 21.1·35-s − 215.·37-s + 119.·39-s + 150.·41-s + 83.0·43-s − 96.5·45-s + 278.·47-s − 325.·49-s − 20.7·51-s − 182.·53-s + 168.·55-s + ⋯ |
| L(s) = 1 | − 0.533·3-s + 0.447·5-s + 0.228·7-s − 0.715·9-s + 0.924·11-s − 0.916·13-s − 0.238·15-s + 0.106·17-s + 0.310·19-s − 0.121·21-s + 0.208·23-s + 0.200·25-s + 0.915·27-s − 0.303·29-s − 0.377·31-s − 0.493·33-s + 0.102·35-s − 0.957·37-s + 0.488·39-s + 0.571·41-s + 0.294·43-s − 0.319·45-s + 0.863·47-s − 0.947·49-s − 0.0570·51-s − 0.472·53-s + 0.413·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 2.77T + 27T^{2} \) |
| 7 | \( 1 - 4.22T + 343T^{2} \) |
| 11 | \( 1 - 33.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.49T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 47.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 65.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 83.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 278.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 182.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 270.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 440.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.07e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 269.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 195.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 136.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 629.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 419.T + 9.12e5T^{2} \) |
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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
−8.658651984570933042072616875819, −7.60362378178508773517405898042, −6.84479690299551896496134151858, −6.00539271498279358544743480076, −5.34088831605711140028094693911, −4.54244723095298727619703873010, −3.39176712738665512180738419409, −2.35314512343187534487665146422, −1.22891382027858587386050653617, 0,
1.22891382027858587386050653617, 2.35314512343187534487665146422, 3.39176712738665512180738419409, 4.54244723095298727619703873010, 5.34088831605711140028094693911, 6.00539271498279358544743480076, 6.84479690299551896496134151858, 7.60362378178508773517405898042, 8.658651984570933042072616875819
