L(s) = 1 | + 2.17·2-s − 3-s + 2.70·4-s + 2.17·5-s − 2.17·6-s + 0.630·7-s + 1.53·8-s + 9-s + 4.70·10-s + 0.290·11-s − 2.70·12-s − 2.07·13-s + 1.36·14-s − 2.17·15-s − 2.07·16-s − 2.34·17-s + 2.17·18-s + 5.07·19-s + 5.87·20-s − 0.630·21-s + 0.630·22-s + 4.17·23-s − 1.53·24-s − 0.290·25-s − 4.51·26-s − 27-s + 1.70·28-s + ⋯ |
L(s) = 1 | + 1.53·2-s − 0.577·3-s + 1.35·4-s + 0.970·5-s − 0.885·6-s + 0.238·7-s + 0.544·8-s + 0.333·9-s + 1.48·10-s + 0.0876·11-s − 0.782·12-s − 0.576·13-s + 0.365·14-s − 0.560·15-s − 0.519·16-s − 0.567·17-s + 0.511·18-s + 1.16·19-s + 1.31·20-s − 0.137·21-s + 0.134·22-s + 0.869·23-s − 0.314·24-s − 0.0581·25-s − 0.884·26-s − 0.192·27-s + 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.681206228\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.681206228\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 7 | \( 1 - 0.630T + 7T^{2} \) |
| 11 | \( 1 - 0.290T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 23 | \( 1 - 4.17T + 23T^{2} \) |
| 29 | \( 1 + 6.04T + 29T^{2} \) |
| 31 | \( 1 + 1.34T + 31T^{2} \) |
| 37 | \( 1 + 6.97T + 37T^{2} \) |
| 41 | \( 1 + 2.78T + 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 + 0.971T + 47T^{2} \) |
| 53 | \( 1 + 1.36T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 4.46T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 - 4.18T + 71T^{2} \) |
| 73 | \( 1 - 0.971T + 73T^{2} \) |
| 79 | \( 1 - 0.447T + 79T^{2} \) |
| 83 | \( 1 - 9.87T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 8.94T + 97T^{2} \) |
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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
−11.81796307359832447310467354896, −11.17941836510278723590781010080, −10.01574983265514802010989734753, −9.067809518568894419264053478798, −7.32440088139654661533892249474, −6.42943883833692173532098943079, −5.42197032920028939187930327597, −4.92164935657430052286963476612, −3.49792210235878334263008152859, −2.03959347967506813760991117114,
2.03959347967506813760991117114, 3.49792210235878334263008152859, 4.92164935657430052286963476612, 5.42197032920028939187930327597, 6.42943883833692173532098943079, 7.32440088139654661533892249474, 9.067809518568894419264053478798, 10.01574983265514802010989734753, 11.17941836510278723590781010080, 11.81796307359832447310467354896