L(s) = 1 | + (−0.642 + 0.766i)2-s + (0.726 + 1.57i)3-s + (−0.173 − 0.984i)4-s + (0.819 + 2.08i)5-s + (−1.67 − 0.454i)6-s + (−4.83 − 0.853i)7-s + (0.866 + 0.500i)8-s + (−1.94 + 2.28i)9-s + (−2.12 − 0.709i)10-s + (−1.60 + 0.584i)11-s + (1.42 − 0.988i)12-s + (1.95 + 2.32i)13-s + (3.76 − 3.15i)14-s + (−2.67 + 2.79i)15-s + (−0.939 + 0.342i)16-s + (3.04 − 1.75i)17-s + ⋯ |
L(s) = 1 | + (−0.454 + 0.541i)2-s + (0.419 + 0.907i)3-s + (−0.0868 − 0.492i)4-s + (0.366 + 0.930i)5-s + (−0.682 − 0.185i)6-s + (−1.82 − 0.322i)7-s + (0.306 + 0.176i)8-s + (−0.648 + 0.761i)9-s + (−0.670 − 0.224i)10-s + (−0.484 + 0.176i)11-s + (0.410 − 0.285i)12-s + (0.541 + 0.645i)13-s + (1.00 − 0.843i)14-s + (−0.691 + 0.722i)15-s + (−0.234 + 0.0855i)16-s + (0.737 − 0.425i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.129094 + 0.840457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.129094 + 0.840457i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.642 - 0.766i)T \) |
| 3 | \( 1 + (-0.726 - 1.57i)T \) |
| 5 | \( 1 + (-0.819 - 2.08i)T \) |
good | 7 | \( 1 + (4.83 + 0.853i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.60 - 0.584i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.95 - 2.32i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.04 + 1.75i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.936 - 1.62i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.79 + 0.492i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.46 - 2.06i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.994 - 5.64i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (1.92 - 1.11i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.451 - 0.379i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.99 + 10.9i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-13.0 - 2.30i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 13.5iT - 53T^{2} \) |
| 59 | \( 1 + (-4.14 - 1.50i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.58 + 8.98i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.47 + 5.32i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.54 - 4.40i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-11.3 - 6.52i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.76 - 3.99i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.47 - 2.94i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (0.280 - 0.485i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.45 + 6.75i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−12.39922256741175069183698719806, −10.81105457644645235429065305723, −10.26430513629738256352160437176, −9.579668357506881822941685183909, −8.766780017708421950931932192291, −7.32476596163475649785402011603, −6.53230949543037763130224444280, −5.47747947673843835871062087288, −3.77824251156729004821370777153, −2.79356539600282242750401218937,
0.71257241544388996553655367028, 2.52125275102307697853183141946, 3.57499871434698862914914039982, 5.64117574902506118725987556024, 6.51949794152109655449244109582, 7.85422177867950558855628721756, 8.725453695591196676027286360162, 9.484121638698805057181820105309, 10.30533568886838468043046168581, 11.77773282273484923826490816592