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 | \( 1 + (-4.40 - 7.62i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.43714694777151284240575854867, −9.557776348973200693256219138972, −9.116097772645567371588763968426, −8.043434044090107013504291384423, −6.91383202655041222235093222516, −5.43425213835938749418423109167, −5.26173157642786761953107896722, −3.56167057514259213955537003699, −2.23961935161897979127902966087, −0.67215294790474298877177235815,
1.51854599648780713647240269983, 3.42179694352292648995554408399, 4.60459595486079229132379207505, 5.93162342332336860876797624784, 6.58555853761653970799052031358, 7.20377968892585560700047576749, 8.279806933108269160065018036667, 9.605779163495130796922181261560, 10.11682514950962295841908483275, 10.81658855692938055253215925723