| L(s) = 1 | − 2-s − 3.03·3-s + 4-s + 1.18·5-s + 3.03·6-s + 7-s − 8-s + 6.20·9-s − 1.18·10-s − 3.03·11-s − 3.03·12-s − 3.36·13-s − 14-s − 3.59·15-s + 16-s − 3.03·17-s − 6.20·18-s + 7.99·19-s + 1.18·20-s − 3.03·21-s + 3.03·22-s + 0.116·23-s + 3.03·24-s − 3.59·25-s + 3.36·26-s − 9.72·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.75·3-s + 0.5·4-s + 0.529·5-s + 1.23·6-s + 0.377·7-s − 0.353·8-s + 2.06·9-s − 0.374·10-s − 0.913·11-s − 0.875·12-s − 0.931·13-s − 0.267·14-s − 0.927·15-s + 0.250·16-s − 0.736·17-s − 1.46·18-s + 1.83·19-s + 0.264·20-s − 0.662·21-s + 0.646·22-s + 0.0243·23-s + 0.619·24-s − 0.719·25-s + 0.659·26-s − 1.87·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4881111221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4881111221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 3.03T + 3T^{2} \) |
| 5 | \( 1 - 1.18T + 5T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 - 7.99T + 19T^{2} \) |
| 23 | \( 1 - 0.116T + 23T^{2} \) |
| 29 | \( 1 + 2.07T + 29T^{2} \) |
| 31 | \( 1 + 9.45T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 7.64T + 43T^{2} \) |
| 47 | \( 1 - 5.07T + 47T^{2} \) |
| 53 | \( 1 - 5.15T + 53T^{2} \) |
| 59 | \( 1 - 8.63T + 59T^{2} \) |
| 61 | \( 1 + 3.51T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 7.88T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 0.500T + 97T^{2} \) |
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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
−7.73774003509852743609905718773, −7.40766617824397624524255822798, −6.67472212311207988711483767403, −5.88536161959115938921912460437, −5.18475965929724654608423910361, −5.02485716373146725700063099802, −3.71908218268528243828431038589, −2.40620690574118485156298265450, −1.57476170081368480646518879646, −0.45300012974377039364205775920,
0.45300012974377039364205775920, 1.57476170081368480646518879646, 2.40620690574118485156298265450, 3.71908218268528243828431038589, 5.02485716373146725700063099802, 5.18475965929724654608423910361, 5.88536161959115938921912460437, 6.67472212311207988711483767403, 7.40766617824397624524255822798, 7.73774003509852743609905718773
