L(s) = 1 | − 2-s − 4-s + (−1 − 2i)5-s + 3·8-s + (1 + 2i)10-s − 2i·11-s + (−3 + 2i)13-s − 16-s − 6i·19-s + (1 + 2i)20-s + 2i·22-s + 6i·23-s + (−3 + 4i)25-s + (3 − 2i)26-s − 6·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.5·4-s + (−0.447 − 0.894i)5-s + 1.06·8-s + (0.316 + 0.632i)10-s − 0.603i·11-s + (−0.832 + 0.554i)13-s − 0.250·16-s − 1.37i·19-s + (0.223 + 0.447i)20-s + 0.426i·22-s + 1.25i·23-s + (−0.600 + 0.800i)25-s + (0.588 − 0.392i)26-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 13 | \( 1 + (3 - 2i)T \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 8iT - 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 2iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 8iT - 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−9.812065024967069423240082930496, −9.302937296842830057253316689864, −8.545183885519234819416679864785, −7.78173344623690426657780822311, −6.86686367727869129681506806680, −5.25701645159624436411223910652, −4.69104779351653812117273012385, −3.45787056510293183704660018165, −1.51982685473719768283489146310, 0,
2.08821501355667929057781753084, 3.58802607574732875031845122592, 4.57621265899631471938526188900, 5.80257991112452127895318376223, 7.11952082237399740368863727939, 7.69806263055736984947067562314, 8.511175411823305983135621210854, 9.628958331813629343406801384610, 10.24708935138090664510373770653