| L(s) = 1 | + (−0.521 + 1.31i)2-s + (1.24 − 1.20i)3-s + (−1.45 − 1.37i)4-s + (3.07 + 1.11i)5-s + (0.940 + 2.26i)6-s + (−1.33 − 0.235i)7-s + (2.56 − 1.19i)8-s + (0.0801 − 2.99i)9-s + (−3.07 + 3.45i)10-s + (−0.982 − 2.69i)11-s + (−3.46 + 0.0555i)12-s + (2.36 + 2.82i)13-s + (1.00 − 1.63i)14-s + (5.16 − 2.32i)15-s + (0.234 + 3.99i)16-s + (1.94 − 1.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.369 + 0.929i)2-s + (0.716 − 0.697i)3-s + (−0.727 − 0.686i)4-s + (1.37 + 0.500i)5-s + (0.383 + 0.923i)6-s + (−0.505 − 0.0891i)7-s + (0.906 − 0.423i)8-s + (0.0267 − 0.999i)9-s + (−0.972 + 1.09i)10-s + (−0.296 − 0.813i)11-s + (−0.999 + 0.0160i)12-s + (0.656 + 0.782i)13-s + (0.269 − 0.437i)14-s + (1.33 − 0.600i)15-s + (0.0587 + 0.998i)16-s + (0.470 − 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34341 + 0.290407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34341 + 0.290407i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.521 - 1.31i)T \) |
| 3 | \( 1 + (-1.24 + 1.20i)T \) |
| good | 5 | \( 1 + (-3.07 - 1.11i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.33 + 0.235i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.982 + 2.69i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.36 - 2.82i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.94 + 1.12i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.22 - 2.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.08 - 6.17i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.00 + 4.19i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (4.65 - 0.820i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (3.70 - 2.14i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.28 - 8.67i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.24 - 1.17i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.21 + 12.5i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 4.14T + 53T^{2} \) |
| 59 | \( 1 + (-3.92 + 10.7i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (9.59 + 1.69i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.64 - 5.57i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.19 - 5.53i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.87 - 4.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.31 - 6.33i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.56 - 1.85i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.06 - 0.612i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-16.9 + 6.17i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.02382725879234693358460718875, −11.26400193858176849759263936520, −9.915487214148755872995060542331, −9.388496134797935250558934577210, −8.382233397745796691869420195344, −7.24187456041037385618111096213, −6.31544268823371598000444794209, −5.66229434094984121581435885454, −3.47999266127715426372743743768, −1.70401834457857213987761300681,
1.92219105726122991102996387828, 3.11580137203212370718251686687, 4.56901336489859274550363911269, 5.70348143385525870764302221210, 7.60186103674911902634019550280, 8.939516358391903263160245180280, 9.262169512985666524633642196284, 10.34965680133658359563800032323, 10.72220646150994377671467894279, 12.66132636964813493972845605974
