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
−12.61658597450042786518150940335, −11.53325106465660566470031693663, −10.69121849853065400162200843332, −9.693485835070928960320303071581, −8.723814692031372697727473753095, −8.092829498495045804097002617423, −6.95169394975746546994009500556, −4.80543825494522275007026849672, −3.64846271331490160132999193163, −1.82046617707922609398549009913,
1.62846660223588520052601682497, 3.23533215741948815079194790916, 5.55736671194473332537811383581, 6.83100103528438224628132193522, 7.80608591119707598715277656064, 8.432930752516202643876007932865, 9.490161762268278590744293091359, 10.70818174496861697700192141711, 11.67825014797114928091304712096, 12.59756890729206513269834659565