| L(s) = 1 | − 2.27·2-s + 1.15·3-s + 3.16·4-s − 5-s − 2.63·6-s − 2.94·7-s − 2.64·8-s − 1.65·9-s + 2.27·10-s + 0.860·11-s + 3.66·12-s + 3.83·13-s + 6.70·14-s − 1.15·15-s − 0.319·16-s − 0.946·17-s + 3.76·18-s − 1.79·19-s − 3.16·20-s − 3.42·21-s − 1.95·22-s + 5.58·23-s − 3.06·24-s + 25-s − 8.70·26-s − 5.39·27-s − 9.33·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 0.669·3-s + 1.58·4-s − 0.447·5-s − 1.07·6-s − 1.11·7-s − 0.934·8-s − 0.551·9-s + 0.718·10-s + 0.259·11-s + 1.05·12-s + 1.06·13-s + 1.79·14-s − 0.299·15-s − 0.0798·16-s − 0.229·17-s + 0.886·18-s − 0.411·19-s − 0.707·20-s − 0.746·21-s − 0.416·22-s + 1.16·23-s − 0.625·24-s + 0.200·25-s − 1.70·26-s − 1.03·27-s − 1.76·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 - 1.15T + 3T^{2} \) |
| 7 | \( 1 + 2.94T + 7T^{2} \) |
| 11 | \( 1 - 0.860T + 11T^{2} \) |
| 13 | \( 1 - 3.83T + 13T^{2} \) |
| 17 | \( 1 + 0.946T + 17T^{2} \) |
| 19 | \( 1 + 1.79T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 - 0.817T + 29T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 - 5.31T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 - 5.48T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 9.07T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 3.16T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 5.09T + 73T^{2} \) |
| 79 | \( 1 + 4.37T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 + 8.17T + 89T^{2} \) |
| 97 | \( 1 + 2.47T + 97T^{2} \) |
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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
−8.892922933602966723714037769468, −8.175290343923592368921799292523, −7.58058218246944356769384946800, −6.55017727680136824621864347768, −6.14146371181239985959449902867, −4.50245862819904669187829915140, −3.31771339770073396290726890982, −2.70479090571202817753569988677, −1.30719773754836255539623499488, 0,
1.30719773754836255539623499488, 2.70479090571202817753569988677, 3.31771339770073396290726890982, 4.50245862819904669187829915140, 6.14146371181239985959449902867, 6.55017727680136824621864347768, 7.58058218246944356769384946800, 8.175290343923592368921799292523, 8.892922933602966723714037769468
