L(s) = 1 | + (−0.5 + 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (1.14 + 1.98i)5-s + (0.5 − 0.866i)6-s + (−1.12 − 2.39i)7-s + 0.999·8-s + 9-s − 2.29·10-s + 0.878·11-s + (0.499 + 0.866i)12-s + (−0.786 + 3.51i)13-s + (2.63 + 0.222i)14-s + (−1.14 − 1.98i)15-s + (−0.5 + 0.866i)16-s + (3.20 + 5.54i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (0.512 + 0.887i)5-s + (0.204 − 0.353i)6-s + (−0.425 − 0.904i)7-s + 0.353·8-s + 0.333·9-s − 0.724·10-s + 0.264·11-s + (0.144 + 0.249i)12-s + (−0.218 + 0.975i)13-s + (0.704 + 0.0593i)14-s + (−0.295 − 0.512i)15-s + (−0.125 + 0.216i)16-s + (0.777 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.488239 + 0.756351i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.488239 + 0.756351i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + (1.12 + 2.39i)T \) |
| 13 | \( 1 + (0.786 - 3.51i)T \) |
good | 5 | \( 1 + (-1.14 - 1.98i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 0.878T + 11T^{2} \) |
| 17 | \( 1 + (-3.20 - 5.54i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 - 1.50T + 19T^{2} \) |
| 23 | \( 1 + (0.658 - 1.14i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.669 + 1.15i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.94 - 3.37i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.69 - 8.12i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.80 - 3.12i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.95 - 8.58i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.188 + 0.327i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.22 - 2.11i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.98 - 5.16i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 4.81T + 61T^{2} \) |
| 67 | \( 1 - 9.75T + 67T^{2} \) |
| 71 | \( 1 + (-1.02 + 1.77i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.432 + 0.749i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.18 + 7.24i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8.66T + 83T^{2} \) |
| 89 | \( 1 + (-6.41 + 11.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.40 + 7.62i)T + (-48.5 - 84.0i)T^{2} \) |
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
−10.81658855692938055253215925723, −10.11682514950962295841908483275, −9.605779163495130796922181261560, −8.279806933108269160065018036667, −7.20377968892585560700047576749, −6.58555853761653970799052031358, −5.93162342332336860876797624784, −4.60459595486079229132379207505, −3.42179694352292648995554408399, −1.51854599648780713647240269983,
0.67215294790474298877177235815, 2.23961935161897979127902966087, 3.56167057514259213955537003699, 5.26173157642786761953107896722, 5.43425213835938749418423109167, 6.91383202655041222235093222516, 8.043434044090107013504291384423, 9.116097772645567371588763968426, 9.557776348973200693256219138972, 10.43714694777151284240575854867