| L(s) = 1 | − 2-s − 1.92·3-s + 4-s − 2.53·5-s + 1.92·6-s − 7-s − 8-s + 0.687·9-s + 2.53·10-s − 3.57·11-s − 1.92·12-s + 3.61·13-s + 14-s + 4.86·15-s + 16-s − 0.257·17-s − 0.687·18-s + 0.598·19-s − 2.53·20-s + 1.92·21-s + 3.57·22-s + 8.37·23-s + 1.92·24-s + 1.40·25-s − 3.61·26-s + 4.44·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.10·3-s + 0.5·4-s − 1.13·5-s + 0.783·6-s − 0.377·7-s − 0.353·8-s + 0.229·9-s + 0.800·10-s − 1.07·11-s − 0.554·12-s + 1.00·13-s + 0.267·14-s + 1.25·15-s + 0.250·16-s − 0.0623·17-s − 0.162·18-s + 0.137·19-s − 0.566·20-s + 0.419·21-s + 0.763·22-s + 1.74·23-s + 0.391·24-s + 0.281·25-s − 0.709·26-s + 0.854·27-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
| \(L(1)\) |
\(\approx\) |
\(0.2153487476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2153487476\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.92T + 3T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 11 | \( 1 + 3.57T + 11T^{2} \) |
| 13 | \( 1 - 3.61T + 13T^{2} \) |
| 17 | \( 1 + 0.257T + 17T^{2} \) |
| 19 | \( 1 - 0.598T + 19T^{2} \) |
| 23 | \( 1 - 8.37T + 23T^{2} \) |
| 29 | \( 1 + 4.10T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 4.11T + 37T^{2} \) |
| 41 | \( 1 - 2.61T + 41T^{2} \) |
| 43 | \( 1 + 2.17T + 43T^{2} \) |
| 47 | \( 1 - 4.93T + 47T^{2} \) |
| 53 | \( 1 - 1.94T + 53T^{2} \) |
| 59 | \( 1 + 8.21T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 8.51T + 67T^{2} \) |
| 71 | \( 1 + 4.42T + 71T^{2} \) |
| 73 | \( 1 + 9.04T + 73T^{2} \) |
| 79 | \( 1 + 2.43T + 79T^{2} \) |
| 83 | \( 1 - 1.85T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.930617549856609236560227131437, −7.40690892711693862447546400333, −6.82184326385492698090029433996, −5.89635575900893733027845106221, −5.42781364713097667433602252537, −4.53960915920288245826475347995, −3.52900098382424429722885788967, −2.88573399828676190843862034196, −1.44058778958727662530046955988, −0.29895669106892863316162302198,
0.29895669106892863316162302198, 1.44058778958727662530046955988, 2.88573399828676190843862034196, 3.52900098382424429722885788967, 4.53960915920288245826475347995, 5.42781364713097667433602252537, 5.89635575900893733027845106221, 6.82184326385492698090029433996, 7.40690892711693862447546400333, 7.930617549856609236560227131437
