| L(s) = 1 | − 0.0264·2-s − 1.87·3-s − 1.99·4-s + 2.52·5-s + 0.0497·6-s − 1.16·7-s + 0.105·8-s + 0.525·9-s − 0.0669·10-s + 0.665·11-s + 3.75·12-s − 13-s + 0.0307·14-s − 4.74·15-s + 3.99·16-s − 0.273·17-s − 0.0139·18-s + 6.34·19-s − 5.05·20-s + 2.18·21-s − 0.0176·22-s − 4.90·23-s − 0.198·24-s + 1.38·25-s + 0.0264·26-s + 4.64·27-s + 2.32·28-s + ⋯ |
| L(s) = 1 | − 0.0187·2-s − 1.08·3-s − 0.999·4-s + 1.13·5-s + 0.0202·6-s − 0.439·7-s + 0.0374·8-s + 0.175·9-s − 0.0211·10-s + 0.200·11-s + 1.08·12-s − 0.277·13-s + 0.00821·14-s − 1.22·15-s + 0.998·16-s − 0.0664·17-s − 0.00328·18-s + 1.45·19-s − 1.12·20-s + 0.475·21-s − 0.00375·22-s − 1.02·23-s − 0.0405·24-s + 0.277·25-s + 0.00519·26-s + 0.894·27-s + 0.438·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0264T + 2T^{2} \) |
| 3 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 + 1.16T + 7T^{2} \) |
| 11 | \( 1 - 0.665T + 11T^{2} \) |
| 17 | \( 1 + 0.273T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 + 0.915T + 31T^{2} \) |
| 37 | \( 1 + 1.13T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 1.90T + 43T^{2} \) |
| 47 | \( 1 - 8.92T + 47T^{2} \) |
| 53 | \( 1 - 2.31T + 53T^{2} \) |
| 59 | \( 1 - 9.78T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 4.99T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 1.68T + 83T^{2} \) |
| 89 | \( 1 - 2.04T + 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.413125763997575783328216821091, −8.633346947095768025581075009396, −7.49269422459537327440773345427, −6.46278060258393594906033926687, −5.63143231805499238435868130817, −5.34184849855315162953527939549, −4.24355781288449965674131593268, −3.03064294270265125496306298236, −1.44724343797129097462993122668, 0,
1.44724343797129097462993122668, 3.03064294270265125496306298236, 4.24355781288449965674131593268, 5.34184849855315162953527939549, 5.63143231805499238435868130817, 6.46278060258393594906033926687, 7.49269422459537327440773345427, 8.633346947095768025581075009396, 9.413125763997575783328216821091
