| L(s) = 1 | − 0.988·2-s + 1.19·3-s − 1.02·4-s − 2.47·5-s − 1.18·6-s + 3.50·7-s + 2.98·8-s − 1.57·9-s + 2.44·10-s + 3.13·11-s − 1.22·12-s − 1.72·13-s − 3.46·14-s − 2.95·15-s − 0.909·16-s − 6.38·17-s + 1.55·18-s − 2.22·19-s + 2.53·20-s + 4.18·21-s − 3.09·22-s − 2.05·23-s + 3.56·24-s + 1.13·25-s + 1.70·26-s − 5.46·27-s − 3.58·28-s + ⋯ |
| L(s) = 1 | − 0.699·2-s + 0.689·3-s − 0.511·4-s − 1.10·5-s − 0.481·6-s + 1.32·7-s + 1.05·8-s − 0.525·9-s + 0.774·10-s + 0.944·11-s − 0.352·12-s − 0.478·13-s − 0.927·14-s − 0.763·15-s − 0.227·16-s − 1.54·17-s + 0.367·18-s − 0.510·19-s + 0.566·20-s + 0.913·21-s − 0.660·22-s − 0.429·23-s + 0.728·24-s + 0.227·25-s + 0.334·26-s − 1.05·27-s − 0.677·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9407326680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9407326680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 - T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 0.988T + 2T^{2} \) |
| 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 2.47T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 + 6.38T + 17T^{2} \) |
| 19 | \( 1 + 2.22T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 - 0.622T + 29T^{2} \) |
| 31 | \( 1 + 1.15T + 31T^{2} \) |
| 37 | \( 1 - 2.31T + 37T^{2} \) |
| 41 | \( 1 + 5.71T + 41T^{2} \) |
| 43 | \( 1 - 6.33T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 8.78T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 67 | \( 1 + 7.33T + 67T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 + 6.72T + 79T^{2} \) |
| 83 | \( 1 - 0.357T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 2.85T + 97T^{2} \) |
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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
−8.088073310586636649402661190101, −7.44473112260527867900682876357, −6.82809456398389403271686617829, −5.66620082163771594689481318065, −4.75565227660934693523558734737, −4.17834336754089509028415848885, −3.78577695768866767545151480066, −2.45430228832958899381651907841, −1.73506836916207029405718773154, −0.51941948187175197164444503422,
0.51941948187175197164444503422, 1.73506836916207029405718773154, 2.45430228832958899381651907841, 3.78577695768866767545151480066, 4.17834336754089509028415848885, 4.75565227660934693523558734737, 5.66620082163771594689481318065, 6.82809456398389403271686617829, 7.44473112260527867900682876357, 8.088073310586636649402661190101
