L(s) = 1 | − 0.470·2-s − 1.77·4-s − 5-s − 7-s + 1.77·8-s + 0.470·10-s − 11-s − 0.249·13-s + 0.470·14-s + 2.71·16-s − 5.71·17-s + 1.83·19-s + 1.77·20-s + 0.470·22-s − 0.778·23-s + 25-s + 0.117·26-s + 1.77·28-s − 3.47·29-s − 3.30·31-s − 4.83·32-s + 2.69·34-s + 35-s + 10.8·37-s − 0.864·38-s − 1.77·40-s − 9.80·41-s + ⋯ |
L(s) = 1 | − 0.332·2-s − 0.889·4-s − 0.447·5-s − 0.377·7-s + 0.628·8-s + 0.148·10-s − 0.301·11-s − 0.0690·13-s + 0.125·14-s + 0.679·16-s − 1.38·17-s + 0.421·19-s + 0.397·20-s + 0.100·22-s − 0.162·23-s + 0.200·25-s + 0.0229·26-s + 0.336·28-s − 0.644·29-s − 0.594·31-s − 0.855·32-s + 0.461·34-s + 0.169·35-s + 1.78·37-s − 0.140·38-s − 0.281·40-s − 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6256194461\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6256194461\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 0.470T + 2T^{2} \) |
| 13 | \( 1 + 0.249T + 13T^{2} \) |
| 17 | \( 1 + 5.71T + 17T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 + 0.778T + 23T^{2} \) |
| 29 | \( 1 + 3.47T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 9.80T + 41T^{2} \) |
| 43 | \( 1 + 5.96T + 43T^{2} \) |
| 47 | \( 1 + 3.30T + 47T^{2} \) |
| 53 | \( 1 - 2.77T + 53T^{2} \) |
| 59 | \( 1 - 3.58T + 59T^{2} \) |
| 61 | \( 1 + 7.27T + 61T^{2} \) |
| 67 | \( 1 - 5.55T + 67T^{2} \) |
| 71 | \( 1 + 2.74T + 71T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 - 7.15T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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
−8.570249818074000345443081150618, −8.005793231616800518922815437614, −7.23299912419115597927769693028, −6.44931315515060495318954311204, −5.44822950880149724318239476558, −4.70693322712190754453895965216, −3.98420859749189257960639120853, −3.14948451412876823396821745959, −1.90647457481864965379221302797, −0.48471064362314672487686158935,
0.48471064362314672487686158935, 1.90647457481864965379221302797, 3.14948451412876823396821745959, 3.98420859749189257960639120853, 4.70693322712190754453895965216, 5.44822950880149724318239476558, 6.44931315515060495318954311204, 7.23299912419115597927769693028, 8.005793231616800518922815437614, 8.570249818074000345443081150618