L(s) = 1 | − 2-s + 2.17·3-s + 4-s + 1.18·5-s − 2.17·6-s − 1.11·7-s − 8-s + 1.70·9-s − 1.18·10-s − 1.35·11-s + 2.17·12-s + 1.19·13-s + 1.11·14-s + 2.57·15-s + 16-s − 2.91·17-s − 1.70·18-s − 6.09·19-s + 1.18·20-s − 2.42·21-s + 1.35·22-s − 2.71·23-s − 2.17·24-s − 3.58·25-s − 1.19·26-s − 2.80·27-s − 1.11·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.25·3-s + 0.5·4-s + 0.531·5-s − 0.885·6-s − 0.422·7-s − 0.353·8-s + 0.569·9-s − 0.375·10-s − 0.408·11-s + 0.626·12-s + 0.332·13-s + 0.298·14-s + 0.665·15-s + 0.250·16-s − 0.707·17-s − 0.402·18-s − 1.39·19-s + 0.265·20-s − 0.529·21-s + 0.288·22-s − 0.565·23-s − 0.442·24-s − 0.717·25-s − 0.235·26-s − 0.538·27-s − 0.211·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
good | 3 | \( 1 - 2.17T + 3T^{2} \) |
| 5 | \( 1 - 1.18T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 17 | \( 1 + 2.91T + 17T^{2} \) |
| 19 | \( 1 + 6.09T + 19T^{2} \) |
| 23 | \( 1 + 2.71T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 37 | \( 1 - 8.60T + 37T^{2} \) |
| 41 | \( 1 - 8.96T + 41T^{2} \) |
| 43 | \( 1 - 3.76T + 43T^{2} \) |
| 47 | \( 1 + 9.20T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 - 6.43T + 67T^{2} \) |
| 71 | \( 1 + 1.35T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 8.70T + 79T^{2} \) |
| 83 | \( 1 + 8.14T + 83T^{2} \) |
| 89 | \( 1 + 8.48T + 89T^{2} \) |
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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
−8.059306491324967941336867971249, −7.23068245785569609642568911330, −6.24388975161861707344977727822, −6.03428153704683246625361729100, −4.63043701529300259119826697194, −3.89136524770810542107441597622, −2.83243732102453850266531270520, −2.41764910630393388419556826761, −1.54641747204695443503296857701, 0,
1.54641747204695443503296857701, 2.41764910630393388419556826761, 2.83243732102453850266531270520, 3.89136524770810542107441597622, 4.63043701529300259119826697194, 6.03428153704683246625361729100, 6.24388975161861707344977727822, 7.23068245785569609642568911330, 8.059306491324967941336867971249