| L(s) = 1 | + 2-s + 0.618·3-s + 4-s − 5-s + 0.618·6-s + 3.23·7-s + 8-s − 2.61·9-s − 10-s + 11-s + 0.618·12-s − 0.618·13-s + 3.23·14-s − 0.618·15-s + 16-s − 4·17-s − 2.61·18-s − 5.61·19-s − 20-s + 2.00·21-s + 22-s + 2·23-s + 0.618·24-s + 25-s − 0.618·26-s − 3.47·27-s + 3.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.356·3-s + 0.5·4-s − 0.447·5-s + 0.252·6-s + 1.22·7-s + 0.353·8-s − 0.872·9-s − 0.316·10-s + 0.301·11-s + 0.178·12-s − 0.171·13-s + 0.864·14-s − 0.159·15-s + 0.250·16-s − 0.970·17-s − 0.617·18-s − 1.28·19-s − 0.223·20-s + 0.436·21-s + 0.213·22-s + 0.417·23-s + 0.126·24-s + 0.200·25-s − 0.121·26-s − 0.668·27-s + 0.611·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 13 | \( 1 + 0.618T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 5.61T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 5.61T + 43T^{2} \) |
| 47 | \( 1 - 3.32T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 + 9.56T + 59T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 4.85T + 71T^{2} \) |
| 79 | \( 1 - 5.70T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 1.52T + 97T^{2} \) |
| 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
−7.47917607531514667084863920723, −6.81224181222548133523481412953, −5.99859161845013325623173131636, −5.28468191125491657550946175621, −4.56809360173540685956859563755, −4.03576340861649591333459269645, −3.16852480933330227162906038976, −2.27943697920290788773864713968, −1.62488475916481178629396135212, 0,
1.62488475916481178629396135212, 2.27943697920290788773864713968, 3.16852480933330227162906038976, 4.03576340861649591333459269645, 4.56809360173540685956859563755, 5.28468191125491657550946175621, 5.99859161845013325623173131636, 6.81224181222548133523481412953, 7.47917607531514667084863920723
