| L(s) = 1 | − 0.649·2-s + 2.92·3-s − 1.57·4-s + 5-s − 1.90·6-s − 7-s + 2.32·8-s + 5.57·9-s − 0.649·10-s − 11-s − 4.62·12-s + 4.48·13-s + 0.649·14-s + 2.92·15-s + 1.64·16-s + 0.928·17-s − 3.62·18-s − 0.252·19-s − 1.57·20-s − 2.92·21-s + 0.649·22-s + 6.27·23-s + 6.80·24-s + 25-s − 2.90·26-s + 7.55·27-s + 1.57·28-s + ⋯ |
| L(s) = 1 | − 0.459·2-s + 1.69·3-s − 0.789·4-s + 0.447·5-s − 0.776·6-s − 0.377·7-s + 0.821·8-s + 1.85·9-s − 0.205·10-s − 0.301·11-s − 1.33·12-s + 1.24·13-s + 0.173·14-s + 0.756·15-s + 0.411·16-s + 0.225·17-s − 0.853·18-s − 0.0579·19-s − 0.352·20-s − 0.639·21-s + 0.138·22-s + 1.30·23-s + 1.38·24-s + 0.200·25-s − 0.570·26-s + 1.45·27-s + 0.298·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.697612672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.697612672\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.649T + 2T^{2} \) |
| 3 | \( 1 - 2.92T + 3T^{2} \) |
| 13 | \( 1 - 4.48T + 13T^{2} \) |
| 17 | \( 1 - 0.928T + 17T^{2} \) |
| 19 | \( 1 + 0.252T + 19T^{2} \) |
| 23 | \( 1 - 6.27T + 23T^{2} \) |
| 29 | \( 1 + 9.10T + 29T^{2} \) |
| 31 | \( 1 + 0.948T + 31T^{2} \) |
| 37 | \( 1 + 0.973T + 37T^{2} \) |
| 41 | \( 1 + 6.20T + 41T^{2} \) |
| 43 | \( 1 + 8.83T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 0.0979T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 2.17T + 73T^{2} \) |
| 79 | \( 1 + 4.38T + 79T^{2} \) |
| 83 | \( 1 + 5.60T + 83T^{2} \) |
| 89 | \( 1 - 1.55T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−10.95572928795414859540244076133, −10.02754698215932977501556045131, −9.248124079841715365324542027428, −8.753836045058276672807118631845, −7.972754285674619701732915163397, −6.97974259761255303147786535319, −5.42085772697015463467650002983, −3.99741632996431172113923498966, −3.12868850339676907247898790943, −1.58078403169662437100881578049,
1.58078403169662437100881578049, 3.12868850339676907247898790943, 3.99741632996431172113923498966, 5.42085772697015463467650002983, 6.97974259761255303147786535319, 7.972754285674619701732915163397, 8.753836045058276672807118631845, 9.248124079841715365324542027428, 10.02754698215932977501556045131, 10.95572928795414859540244076133
