L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s − 4·13-s + 16-s − 4·17-s − 18-s + 19-s − 22-s + 5·23-s + 24-s + 4·26-s − 27-s + 3·29-s + 5·31-s − 32-s − 33-s + 4·34-s + 36-s − 6·37-s − 38-s + 4·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.229·19-s − 0.213·22-s + 1.04·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.174·33-s + 0.685·34-s + 1/6·36-s − 0.986·37-s − 0.162·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−13.65284037971889, −13.07694945373585, −12.47771660439660, −12.21203057264144, −11.62817384203285, −11.25216498431186, −10.76214147741078, −10.23546046413976, −9.865292138719269, −9.314567157025445, −8.836664613647359, −8.412234441494663, −7.717308780654727, −7.232392067457597, −6.761356241124328, −6.479342889972456, −5.718583267435044, −5.124947497501591, −4.758522654505042, −4.070039929450376, −3.429029796666542, −2.489262436300247, −2.361883256061936, −1.328507919933219, −0.7758468468194496, 0,
0.7758468468194496, 1.328507919933219, 2.361883256061936, 2.489262436300247, 3.429029796666542, 4.070039929450376, 4.758522654505042, 5.124947497501591, 5.718583267435044, 6.479342889972456, 6.761356241124328, 7.232392067457597, 7.717308780654727, 8.412234441494663, 8.836664613647359, 9.314567157025445, 9.865292138719269, 10.23546046413976, 10.76214147741078, 11.25216498431186, 11.62817384203285, 12.21203057264144, 12.47771660439660, 13.07694945373585, 13.65284037971889