L(s) = 1 | + 5-s − 3·7-s + 11-s + 6·17-s − 5·23-s + 25-s − 5·29-s − 3·31-s − 3·35-s + 6·37-s + 7·41-s + 5·43-s − 3·47-s + 2·49-s + 9·53-s + 55-s − 12·59-s + 12·61-s + 16·67-s − 10·71-s + 13·73-s − 3·77-s − 6·79-s − 4·83-s + 6·85-s + 14·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.13·7-s + 0.301·11-s + 1.45·17-s − 1.04·23-s + 1/5·25-s − 0.928·29-s − 0.538·31-s − 0.507·35-s + 0.986·37-s + 1.09·41-s + 0.762·43-s − 0.437·47-s + 2/7·49-s + 1.23·53-s + 0.134·55-s − 1.56·59-s + 1.53·61-s + 1.95·67-s − 1.18·71-s + 1.52·73-s − 0.341·77-s − 0.675·79-s − 0.439·83-s + 0.650·85-s + 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−12.84130373749846, −12.45622905902397, −12.02041650137654, −11.38613412269728, −11.09525918332902, −10.36305714842144, −9.994975994156153, −9.686761303601740, −9.285944510002822, −8.846864887198838, −8.159927850356693, −7.686776149604252, −7.321746572816187, −6.721352061196980, −6.154593130727390, −5.897387047371050, −5.456483260593748, −4.861432575839294, −4.045707728127339, −3.735059475234954, −3.277894620208403, −2.544063420210033, −2.189464821547997, −1.335972206292006, −0.7971987000093110, 0,
0.7971987000093110, 1.335972206292006, 2.189464821547997, 2.544063420210033, 3.277894620208403, 3.735059475234954, 4.045707728127339, 4.861432575839294, 5.456483260593748, 5.897387047371050, 6.154593130727390, 6.721352061196980, 7.321746572816187, 7.686776149604252, 8.159927850356693, 8.846864887198838, 9.285944510002822, 9.686761303601740, 9.994975994156153, 10.36305714842144, 11.09525918332902, 11.38613412269728, 12.02041650137654, 12.45622905902397, 12.84130373749846