| L(s) = 1 | − 9.87·3-s + 16.8·5-s + 5.21·7-s + 70.6·9-s + 8.55·11-s + 11.3·13-s − 166.·15-s + 68.4·17-s − 6.93·19-s − 51.5·21-s − 132.·23-s + 157.·25-s − 430.·27-s + 29·29-s − 0.419·31-s − 84.4·33-s + 87.8·35-s − 395.·37-s − 112.·39-s − 447.·41-s − 184.·43-s + 1.18e3·45-s − 97.2·47-s − 315.·49-s − 676.·51-s + 209.·53-s + 143.·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 1.50·5-s + 0.281·7-s + 2.61·9-s + 0.234·11-s + 0.241·13-s − 2.86·15-s + 0.976·17-s − 0.0836·19-s − 0.535·21-s − 1.19·23-s + 1.26·25-s − 3.07·27-s + 0.185·29-s − 0.00242·31-s − 0.445·33-s + 0.424·35-s − 1.75·37-s − 0.460·39-s − 1.70·41-s − 0.653·43-s + 3.93·45-s − 0.301·47-s − 0.920·49-s − 1.85·51-s + 0.543·53-s + 0.352·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 9.87T + 27T^{2} \) |
| 5 | \( 1 - 16.8T + 125T^{2} \) |
| 7 | \( 1 - 5.21T + 343T^{2} \) |
| 11 | \( 1 - 8.55T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.93T + 6.85e3T^{2} \) |
| 23 | \( 1 + 132.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 0.419T + 2.97e4T^{2} \) |
| 37 | \( 1 + 395.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 447.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 184.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 97.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 209.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 45.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 427.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 405.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 557.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 381.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 577.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 353.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 677.T + 9.12e5T^{2} \) |
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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
−8.566540692704512765023375134665, −7.39451341743459973178056683761, −6.49484031661583710810281065130, −6.08591004408998976445317165166, −5.32049357261464651929409068216, −4.86496192334696858442598665179, −3.60369921260683542776789615104, −1.86844166743318844357118936195, −1.31372328702898687549741934268, 0,
1.31372328702898687549741934268, 1.86844166743318844357118936195, 3.60369921260683542776789615104, 4.86496192334696858442598665179, 5.32049357261464651929409068216, 6.08591004408998976445317165166, 6.49484031661583710810281065130, 7.39451341743459973178056683761, 8.566540692704512765023375134665
