| L(s) = 1 | + 0.0482·2-s − 3-s − 1.99·4-s + 1.12·5-s − 0.0482·6-s − 3.19·7-s − 0.192·8-s + 9-s + 0.0544·10-s − 5.55·11-s + 1.99·12-s − 2.12·13-s − 0.154·14-s − 1.12·15-s + 3.98·16-s + 17-s + 0.0482·18-s + 4.53·19-s − 2.25·20-s + 3.19·21-s − 0.268·22-s − 6.46·23-s + 0.192·24-s − 3.72·25-s − 0.102·26-s − 27-s + 6.37·28-s + ⋯ |
| L(s) = 1 | + 0.0341·2-s − 0.577·3-s − 0.998·4-s + 0.504·5-s − 0.0197·6-s − 1.20·7-s − 0.0682·8-s + 0.333·9-s + 0.0172·10-s − 1.67·11-s + 0.576·12-s − 0.590·13-s − 0.0411·14-s − 0.291·15-s + 0.996·16-s + 0.242·17-s + 0.0113·18-s + 1.04·19-s − 0.503·20-s + 0.696·21-s − 0.0572·22-s − 1.34·23-s + 0.0393·24-s − 0.745·25-s − 0.0201·26-s − 0.192·27-s + 1.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09475265022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09475265022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0482T + 2T^{2} \) |
| 5 | \( 1 - 1.12T + 5T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 + 5.55T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 23 | \( 1 + 6.46T + 23T^{2} \) |
| 29 | \( 1 + 7.35T + 29T^{2} \) |
| 31 | \( 1 + 3.74T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 + 7.26T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 7.51T + 53T^{2} \) |
| 59 | \( 1 - 3.99T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 + 9.30T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 7.54T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 3.37T + 97T^{2} \) |
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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
−7.66382856548147417728221847130, −7.33692228827829797050485279440, −6.12190798917226499499099209626, −5.69579524635522777200175673078, −5.23499706068833909606594183968, −4.39257170316652894444061163038, −3.50991272393084716871643329706, −2.81644355484300106517869078005, −1.69791775651806520003424820636, −0.15145881280430682004481781378,
0.15145881280430682004481781378, 1.69791775651806520003424820636, 2.81644355484300106517869078005, 3.50991272393084716871643329706, 4.39257170316652894444061163038, 5.23499706068833909606594183968, 5.69579524635522777200175673078, 6.12190798917226499499099209626, 7.33692228827829797050485279440, 7.66382856548147417728221847130
