| L(s) = 1 | − 2-s − 3.25·3-s + 4-s − 0.797·5-s + 3.25·6-s − 4.64·7-s − 8-s + 7.59·9-s + 0.797·10-s + 3.29·11-s − 3.25·12-s − 5.46·13-s + 4.64·14-s + 2.59·15-s + 16-s + 0.500·17-s − 7.59·18-s + 1.04·19-s − 0.797·20-s + 15.1·21-s − 3.29·22-s − 3.58·23-s + 3.25·24-s − 4.36·25-s + 5.46·26-s − 14.9·27-s − 4.64·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.87·3-s + 0.5·4-s − 0.356·5-s + 1.32·6-s − 1.75·7-s − 0.353·8-s + 2.53·9-s + 0.252·10-s + 0.992·11-s − 0.939·12-s − 1.51·13-s + 1.24·14-s + 0.670·15-s + 0.250·16-s + 0.121·17-s − 1.79·18-s + 0.239·19-s − 0.178·20-s + 3.29·21-s − 0.701·22-s − 0.746·23-s + 0.664·24-s − 0.872·25-s + 1.07·26-s − 2.87·27-s − 0.877·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 + 0.797T + 5T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 11 | \( 1 - 3.29T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 0.500T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 - 8.16T + 31T^{2} \) |
| 37 | \( 1 - 0.468T + 37T^{2} \) |
| 41 | \( 1 + 0.438T + 41T^{2} \) |
| 43 | \( 1 - 9.00T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 - 1.85T + 53T^{2} \) |
| 59 | \( 1 + 3.16T + 59T^{2} \) |
| 61 | \( 1 + 9.13T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 + 2.40T + 71T^{2} \) |
| 73 | \( 1 - 5.41T + 73T^{2} \) |
| 79 | \( 1 - 17.6T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 5.40T + 89T^{2} \) |
| 97 | \( 1 - 8.63T + 97T^{2} \) |
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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
−7.69566838573957957456942422966, −7.28225058224133971223750858789, −6.40741126035131907223485595783, −6.20543634993722801378008718114, −5.32954352163376006797727896579, −4.30709738212544399281189311455, −3.56487963256700823918714499684, −2.21421086086731631771522583129, −0.790608843103161693838901403435, 0,
0.790608843103161693838901403435, 2.21421086086731631771522583129, 3.56487963256700823918714499684, 4.30709738212544399281189311455, 5.32954352163376006797727896579, 6.20543634993722801378008718114, 6.40741126035131907223485595783, 7.28225058224133971223750858789, 7.69566838573957957456942422966
