| L(s) = 1 | − 1.73·2-s − 3-s + 0.999·4-s − 5-s + 1.73·6-s − 1.26·7-s + 1.73·8-s + 9-s + 1.73·10-s + 5.46·11-s − 0.999·12-s − 4.73·13-s + 2.19·14-s + 15-s − 5·16-s + 5.46·17-s − 1.73·18-s − 1.46·19-s − 0.999·20-s + 1.26·21-s − 9.46·22-s − 7.46·23-s − 1.73·24-s + 25-s + 8.19·26-s − 27-s − 1.26·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 0.577·3-s + 0.499·4-s − 0.447·5-s + 0.707·6-s − 0.479·7-s + 0.612·8-s + 0.333·9-s + 0.547·10-s + 1.64·11-s − 0.288·12-s − 1.31·13-s + 0.586·14-s + 0.258·15-s − 1.25·16-s + 1.32·17-s − 0.408·18-s − 0.335·19-s − 0.223·20-s + 0.276·21-s − 2.01·22-s − 1.55·23-s − 0.353·24-s + 0.200·25-s + 1.60·26-s − 0.192·27-s − 0.239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 4.73T + 13T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 + 7.46T + 23T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 37 | \( 1 + 3.26T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 + 1.80T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 0.196T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.15804034468201081491465804556, −9.854586141575514782469273280567, −8.931787304260799311407262609655, −7.88154516517883126710708604625, −7.11969238303714121053718129153, −6.18704710676757480373093682864, −4.77544166930372631864507208177, −3.65319233184203824627886448868, −1.61937606310933530196373578619, 0,
1.61937606310933530196373578619, 3.65319233184203824627886448868, 4.77544166930372631864507208177, 6.18704710676757480373093682864, 7.11969238303714121053718129153, 7.88154516517883126710708604625, 8.931787304260799311407262609655, 9.854586141575514782469273280567, 10.15804034468201081491465804556
