L(s) = 1 | + (−0.651 + 1.12i)2-s + (−0.5 + 0.866i)3-s + (0.151 + 0.262i)4-s − 5-s + (−0.651 − 1.12i)6-s + (0.5 + 0.866i)7-s − 3·8-s + (1 + 1.73i)9-s + (0.651 − 1.12i)10-s + (2.80 − 4.85i)11-s − 0.302·12-s + 3.60·13-s − 1.30·14-s + (0.5 − 0.866i)15-s + (1.65 − 2.86i)16-s + (−0.197 − 0.341i)17-s + ⋯ |
L(s) = 1 | + (−0.460 + 0.797i)2-s + (−0.288 + 0.499i)3-s + (0.0756 + 0.131i)4-s − 0.447·5-s + (−0.265 − 0.460i)6-s + (0.188 + 0.327i)7-s − 1.06·8-s + (0.333 + 0.577i)9-s + (0.205 − 0.356i)10-s + (0.845 − 1.46i)11-s − 0.0874·12-s + 1.00·13-s − 0.348·14-s + (0.129 − 0.223i)15-s + (0.412 − 0.715i)16-s + (−0.0478 − 0.0828i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.420340 + 0.544176i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.420340 + 0.544176i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 13 | \( 1 - 3.60T \) |
good | 2 | \( 1 + (0.651 - 1.12i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.80 + 4.85i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (0.197 + 0.341i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.802 + 1.39i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.10 - 7.11i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-1.80 + 3.12i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.10 + 3.64i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + (5.40 + 9.36i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.40 - 14.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 9.21T + 79T^{2} \) |
| 83 | \( 1 - 5.21T + 83T^{2} \) |
| 89 | \( 1 + (4.10 - 7.11i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.80 - 13.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−15.57877162505942249857984663148, −14.49399711499434791262040728068, −13.06925542886101611705099764002, −11.58835307892041387538734409139, −10.87905492944601137255752967700, −9.036532808057901851766936685973, −8.308002667946383299678719688113, −6.87919402287606897317926620573, −5.57902814091702612338913124920, −3.63455688525051791274289816000,
1.51545986592200855846070605594, 3.95016738525998569579110113141, 6.18456918179466007260633386603, 7.36207989365010637180779909424, 9.078392531186914942929991916857, 10.09638830681397758136885600406, 11.38273753470751407937937588343, 12.04116950388572674951618047530, 13.10589483965291201465374908671, 14.77897557482352577259779981659