| L(s) = 1 | − 3.82·2-s − 2·3-s + 6.65·4-s + 6.48·5-s + 7.65·6-s + 7·7-s + 5.14·8-s − 23·9-s − 24.8·10-s − 11·11-s − 13.3·12-s − 45.6·13-s − 26.7·14-s − 12.9·15-s − 72.9·16-s − 63.6·17-s + 88.0·18-s − 39.5·19-s + 43.1·20-s − 14·21-s + 42.1·22-s − 78.9·23-s − 10.2·24-s − 82.9·25-s + 174.·26-s + 100·27-s + 46.5·28-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 0.384·3-s + 0.832·4-s + 0.580·5-s + 0.520·6-s + 0.377·7-s + 0.227·8-s − 0.851·9-s − 0.785·10-s − 0.301·11-s − 0.320·12-s − 0.974·13-s − 0.511·14-s − 0.223·15-s − 1.13·16-s − 0.908·17-s + 1.15·18-s − 0.477·19-s + 0.482·20-s − 0.145·21-s + 0.408·22-s − 0.715·23-s − 0.0874·24-s − 0.663·25-s + 1.31·26-s + 0.712·27-s + 0.314·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 3.82T + 8T^{2} \) |
| 3 | \( 1 + 2T + 27T^{2} \) |
| 5 | \( 1 - 6.48T + 125T^{2} \) |
| 13 | \( 1 + 45.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 63.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 39.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 256.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 170.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 307.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 316T + 7.95e4T^{2} \) |
| 47 | \( 1 + 576.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 173.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 82.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 63.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 349.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 119.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 573.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 568.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 510.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 396.T + 9.12e5T^{2} \) |
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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
−13.49964337600790644751417300858, −11.98662224767615646659283975413, −10.89120689087137955857271315956, −10.01230673665230531288734432945, −8.903413153066784212705311550277, −7.916843754645860342332527807039, −6.48897275377878676500790445963, −4.94065500614726169106267074673, −2.14749961162676219477459569272, 0,
2.14749961162676219477459569272, 4.94065500614726169106267074673, 6.48897275377878676500790445963, 7.916843754645860342332527807039, 8.903413153066784212705311550277, 10.01230673665230531288734432945, 10.89120689087137955857271315956, 11.98662224767615646659283975413, 13.49964337600790644751417300858
