| L(s) = 1 | − 2-s + 0.648·3-s + 4-s − 3.70·5-s − 0.648·6-s + 0.325·7-s − 8-s − 2.57·9-s + 3.70·10-s + 1.12·11-s + 0.648·12-s − 1.48·13-s − 0.325·14-s − 2.40·15-s + 16-s + 1.52·17-s + 2.57·18-s − 3.07·19-s − 3.70·20-s + 0.210·21-s − 1.12·22-s + 5.81·23-s − 0.648·24-s + 8.75·25-s + 1.48·26-s − 3.61·27-s + 0.325·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.374·3-s + 0.5·4-s − 1.65·5-s − 0.264·6-s + 0.122·7-s − 0.353·8-s − 0.859·9-s + 1.17·10-s + 0.339·11-s + 0.187·12-s − 0.412·13-s − 0.0869·14-s − 0.620·15-s + 0.250·16-s + 0.370·17-s + 0.608·18-s − 0.705·19-s − 0.829·20-s + 0.0460·21-s − 0.240·22-s + 1.21·23-s − 0.132·24-s + 1.75·25-s + 0.291·26-s − 0.696·27-s + 0.0614·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 0.648T + 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 - 0.325T + 7T^{2} \) |
| 11 | \( 1 - 1.12T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 - 5.81T + 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 - 9.20T + 31T^{2} \) |
| 37 | \( 1 - 0.515T + 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 - 8.01T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 - 3.51T + 59T^{2} \) |
| 61 | \( 1 + 7.44T + 61T^{2} \) |
| 67 | \( 1 + 4.46T + 67T^{2} \) |
| 71 | \( 1 + 7.54T + 71T^{2} \) |
| 73 | \( 1 + 8.07T + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−8.229747051237457071714823627107, −7.55209210151615241758296627561, −6.88661152680316933514372179394, −6.08448220685528005677162132511, −4.88190589574769114542894641956, −4.21400538669027261845200523214, −3.16079130105969909948871632403, −2.69748195979737031700467705299, −1.12275659839465808517483198685, 0,
1.12275659839465808517483198685, 2.69748195979737031700467705299, 3.16079130105969909948871632403, 4.21400538669027261845200523214, 4.88190589574769114542894641956, 6.08448220685528005677162132511, 6.88661152680316933514372179394, 7.55209210151615241758296627561, 8.229747051237457071714823627107
