| L(s) = 1 | + 1.73·2-s + 3-s + 0.999·4-s + 1.73·5-s + 1.73·6-s − 1.73·8-s − 2·9-s + 2.99·10-s − 5.19·11-s + 0.999·12-s + 1.73·15-s − 5·16-s + 6·17-s − 3.46·18-s + 1.73·19-s + 1.73·20-s − 9·22-s − 1.73·24-s − 2.00·25-s − 5·27-s + 3·29-s + 2.99·30-s + 1.73·31-s − 5.19·32-s − 5.19·33-s + 10.3·34-s − 1.99·36-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.499·4-s + 0.774·5-s + 0.707·6-s − 0.612·8-s − 0.666·9-s + 0.948·10-s − 1.56·11-s + 0.288·12-s + 0.447·15-s − 1.25·16-s + 1.45·17-s − 0.816·18-s + 0.397·19-s + 0.387·20-s − 1.91·22-s − 0.353·24-s − 0.400·25-s − 0.962·27-s + 0.557·29-s + 0.547·30-s + 0.311·31-s − 0.918·32-s − 0.904·33-s + 1.78·34-s − 0.333·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 + 1.73T + 71T^{2} \) |
| 73 | \( 1 - 8.66T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
| 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
−7.58071904157134040307153138615, −6.39706357306443940817179074327, −5.90680971072402773235605787109, −5.22618926011741987322024888966, −4.92014687631598945123702884018, −3.76488741181321131598100946873, −2.95709742480635951174909643686, −2.75956247429299942299187588274, −1.65147260684701594983488640076, 0,
1.65147260684701594983488640076, 2.75956247429299942299187588274, 2.95709742480635951174909643686, 3.76488741181321131598100946873, 4.92014687631598945123702884018, 5.22618926011741987322024888966, 5.90680971072402773235605787109, 6.39706357306443940817179074327, 7.58071904157134040307153138615
