L(s) = 1 | + (−1.5 + 0.866i)3-s + (−1.63 + 1.52i)5-s + (−2.63 + 0.209i)7-s + (−1.13 − 1.97i)11-s + 6.09i·13-s + (1.13 − 3.70i)15-s + (4.13 − 2.38i)17-s + (−2.13 + 3.70i)19-s + (3.77 − 2.59i)21-s + (0.774 + 0.447i)23-s + (0.362 − 4.98i)25-s − 5.19i·27-s + 3.27·29-s + (2.13 + 3.70i)31-s + (3.41 + 1.97i)33-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)3-s + (−0.732 + 0.680i)5-s + (−0.996 + 0.0791i)7-s + (−0.342 − 0.594i)11-s + 1.68i·13-s + (0.293 − 0.955i)15-s + (1.00 − 0.579i)17-s + (−0.490 + 0.849i)19-s + (0.823 − 0.566i)21-s + (0.161 + 0.0932i)23-s + (0.0725 − 0.997i)25-s − 0.999i·27-s + 0.608·29-s + (0.383 + 0.664i)31-s + (0.594 + 0.342i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.151385 + 0.428563i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.151385 + 0.428563i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.63 - 1.52i)T \) |
| 7 | \( 1 + (2.63 - 0.209i)T \) |
good | 3 | \( 1 + (1.5 - 0.866i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.13 + 1.97i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6.09iT - 13T^{2} \) |
| 17 | \( 1 + (-4.13 + 2.38i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.13 - 3.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.774 - 0.447i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.27T + 29T^{2} \) |
| 31 | \( 1 + (-2.13 - 3.70i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.86 - 2.80i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 6.50iT - 43T^{2} \) |
| 47 | \( 1 + (1.86 + 1.07i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.41 - 3.70i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.13 - 3.70i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.774 - 1.34i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12.0 - 6.95i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + (1.86 - 1.07i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.137 - 0.238i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.67iT - 83T^{2} \) |
| 89 | \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−13.63353552183336379371438203800, −12.13564299656920956143222213114, −11.58346545675203285839844490495, −10.55915831480658510633097737607, −9.742130408627875239713683139104, −8.286320171023992267639999722613, −6.88658901016244762549124394065, −5.99148027870127104092094161410, −4.50145419107119440217555203934, −3.13998901436074136924481282816,
0.50217043291271349858529460060, 3.34268727866169124804183668448, 5.04808688066413499244962933276, 6.13039109344885978473484787499, 7.33544246128321898874445264117, 8.394681616871877058903798514481, 9.802873927752032924063979513318, 10.81921208898192317672459429110, 12.05797152275399847413047340096, 12.65273153729242921623148193437