L(s) = 1 | + (−1.32 + 0.481i)2-s + (1.28 − 1.16i)3-s + (1.53 − 1.27i)4-s + (−0.646 + 1.11i)5-s + (−1.14 + 2.16i)6-s + (2.42 + 1.05i)7-s + (−1.42 + 2.44i)8-s + (0.300 − 2.98i)9-s + (0.321 − 1.79i)10-s + (1.60 − 0.923i)11-s + (0.488 − 3.42i)12-s − 2.25i·13-s + (−3.73 − 0.232i)14-s + (0.470 + 2.18i)15-s + (0.726 − 3.93i)16-s + (3.89 − 2.24i)17-s + ⋯ |
L(s) = 1 | + (−0.940 + 0.340i)2-s + (0.741 − 0.670i)3-s + (0.768 − 0.639i)4-s + (−0.289 + 0.500i)5-s + (−0.469 + 0.883i)6-s + (0.917 + 0.397i)7-s + (−0.505 + 0.862i)8-s + (0.100 − 0.994i)9-s + (0.101 − 0.569i)10-s + (0.482 − 0.278i)11-s + (0.140 − 0.990i)12-s − 0.625i·13-s + (−0.998 − 0.0620i)14-s + (0.121 + 0.565i)15-s + (0.181 − 0.983i)16-s + (0.944 − 0.545i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00786 - 0.0707345i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00786 - 0.0707345i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.32 - 0.481i)T \) |
| 3 | \( 1 + (-1.28 + 1.16i)T \) |
| 7 | \( 1 + (-2.42 - 1.05i)T \) |
good | 5 | \( 1 + (0.646 - 1.11i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.60 + 0.923i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.25iT - 13T^{2} \) |
| 17 | \( 1 + (-3.89 + 2.24i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.80 - 4.86i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.519 + 0.900i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.32T + 29T^{2} \) |
| 31 | \( 1 + (3.69 - 2.13i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (8.18 + 4.72i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.39iT - 41T^{2} \) |
| 43 | \( 1 + 6.02T + 43T^{2} \) |
| 47 | \( 1 + (5.90 - 10.2i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.02 + 10.4i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.57 - 5.52i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.65 - 4.41i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.05 - 5.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + (4.38 + 7.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.37 - 1.36i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.74iT - 83T^{2} \) |
| 89 | \( 1 + (8.31 + 4.79i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8.73T + 97T^{2} \) |
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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
−12.59756890729206513269834659565, −11.67825014797114928091304712096, −10.70818174496861697700192141711, −9.490161762268278590744293091359, −8.432930752516202643876007932865, −7.80608591119707598715277656064, −6.83100103528438224628132193522, −5.55736671194473332537811383581, −3.23533215741948815079194790916, −1.62846660223588520052601682497,
1.82046617707922609398549009913, 3.64846271331490160132999193163, 4.80543825494522275007026849672, 6.95169394975746546994009500556, 8.092829498495045804097002617423, 8.723814692031372697727473753095, 9.693485835070928960320303071581, 10.69121849853065400162200843332, 11.53325106465660566470031693663, 12.61658597450042786518150940335