| L(s) = 1 | + 2-s − 1.48·3-s + 4-s + 1.68·5-s − 1.48·6-s + 3.31·7-s + 8-s − 0.789·9-s + 1.68·10-s − 2.31·11-s − 1.48·12-s + 3.30·13-s + 3.31·14-s − 2.50·15-s + 16-s − 0.160·17-s − 0.789·18-s − 7.08·19-s + 1.68·20-s − 4.92·21-s − 2.31·22-s − 5.83·23-s − 1.48·24-s − 2.15·25-s + 3.30·26-s + 5.63·27-s + 3.31·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.858·3-s + 0.5·4-s + 0.754·5-s − 0.606·6-s + 1.25·7-s + 0.353·8-s − 0.263·9-s + 0.533·10-s − 0.698·11-s − 0.429·12-s + 0.916·13-s + 0.884·14-s − 0.647·15-s + 0.250·16-s − 0.0389·17-s − 0.186·18-s − 1.62·19-s + 0.377·20-s − 1.07·21-s − 0.494·22-s − 1.21·23-s − 0.303·24-s − 0.430·25-s + 0.647·26-s + 1.08·27-s + 0.625·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 1.48T + 3T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 + 2.31T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 + 0.160T + 17T^{2} \) |
| 19 | \( 1 + 7.08T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 + 4.95T + 29T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 0.999T + 41T^{2} \) |
| 43 | \( 1 + 4.21T + 43T^{2} \) |
| 47 | \( 1 + 1.35T + 47T^{2} \) |
| 53 | \( 1 + 4.42T + 53T^{2} \) |
| 59 | \( 1 + 6.99T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.73317074580788503637515016571, −6.69354238515453278267663203054, −6.01492834854140372885312437379, −5.65334259623204935839091326417, −4.95106741513374514052697507842, −4.30581832821192374305222275689, −3.34048272857827445669050419879, −2.06720059144447940499798950382, −1.69458314274102506286390405793, 0,
1.69458314274102506286390405793, 2.06720059144447940499798950382, 3.34048272857827445669050419879, 4.30581832821192374305222275689, 4.95106741513374514052697507842, 5.65334259623204935839091326417, 6.01492834854140372885312437379, 6.69354238515453278267663203054, 7.73317074580788503637515016571
