| L(s) = 1 | + 2-s + 4-s + 2·5-s − 2·7-s + 8-s + 2·10-s + 11-s + 13-s − 2·14-s + 16-s + 17-s − 4·19-s + 2·20-s + 22-s + 4·23-s − 25-s + 26-s − 2·28-s − 2·29-s + 32-s + 34-s − 4·35-s − 2·37-s − 4·38-s + 2·40-s + 6·41-s − 12·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.353·8-s + 0.632·10-s + 0.301·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.447·20-s + 0.213·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.377·28-s − 0.371·29-s + 0.176·32-s + 0.171·34-s − 0.676·35-s − 0.328·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s − 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.87294890731801, −14.43729356175234, −13.75092179725315, −13.40052616349697, −12.88806408811607, −12.66875871145566, −11.79272881233776, −11.47744975160299, −10.71230422108837, −10.25571273756314, −9.814473766557765, −9.119902455790110, −8.791411209962726, −7.922517287074153, −7.345360287192409, −6.581824087386233, −6.273413750714709, −5.878638943204026, −5.024425988202995, −4.686235084740050, −3.656885879959240, −3.400163795631998, −2.543544623447966, −1.919880118332879, −1.214167710490007, 0,
1.214167710490007, 1.919880118332879, 2.543544623447966, 3.400163795631998, 3.656885879959240, 4.686235084740050, 5.024425988202995, 5.878638943204026, 6.273413750714709, 6.581824087386233, 7.345360287192409, 7.922517287074153, 8.791411209962726, 9.119902455790110, 9.814473766557765, 10.25571273756314, 10.71230422108837, 11.47744975160299, 11.79272881233776, 12.66875871145566, 12.88806408811607, 13.40052616349697, 13.75092179725315, 14.43729356175234, 14.87294890731801
