| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 7.67·5-s + 6·6-s + 7.46·7-s − 8·8-s + 9·9-s + 15.3·10-s − 35.6·11-s − 12·12-s + 66.8·13-s − 14.9·14-s + 23.0·15-s + 16·16-s + 23.7·17-s − 18·18-s + 16.2·19-s − 30.7·20-s − 22.4·21-s + 71.2·22-s − 86.5·23-s + 24·24-s − 66.0·25-s − 133.·26-s − 27·27-s + 29.8·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.686·5-s + 0.408·6-s + 0.403·7-s − 0.353·8-s + 0.333·9-s + 0.485·10-s − 0.976·11-s − 0.288·12-s + 1.42·13-s − 0.285·14-s + 0.396·15-s + 0.250·16-s + 0.339·17-s − 0.235·18-s + 0.196·19-s − 0.343·20-s − 0.232·21-s + 0.690·22-s − 0.784·23-s + 0.204·24-s − 0.528·25-s − 1.00·26-s − 0.192·27-s + 0.201·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 59 | \( 1 - 59T \) |
| good | 5 | \( 1 + 7.67T + 125T^{2} \) |
| 7 | \( 1 - 7.46T + 343T^{2} \) |
| 11 | \( 1 + 35.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 86.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 251.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 76.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 406.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 528.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 696.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 307.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 812.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 425.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 502.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 658.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 527.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.01e3T + 9.12e5T^{2} \) |
| 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
−10.55628700926631173088132953365, −9.891557482851237944668454270425, −8.309351346066967166235993015522, −8.109420009485605280298938348940, −6.80800228371007767715727959265, −5.82191687295601332768694171268, −4.59779798168936298448023389474, −3.21713143614586783637356097973, −1.43820514846492083335553541529, 0,
1.43820514846492083335553541529, 3.21713143614586783637356097973, 4.59779798168936298448023389474, 5.82191687295601332768694171268, 6.80800228371007767715727959265, 8.109420009485605280298938348940, 8.309351346066967166235993015522, 9.891557482851237944668454270425, 10.55628700926631173088132953365
