L(s) = 1 | + (−0.447 + 1.66i)3-s + (0.849 + 2.06i)5-s + (−2.48 − 0.900i)7-s + (0.0138 + 0.00800i)9-s + (1.10 + 1.91i)11-s + (1.91 + 1.91i)13-s + (−3.83 + 0.491i)15-s + (−4.19 − 1.12i)17-s + (−1.80 + 3.12i)19-s + (2.61 − 3.74i)21-s + (0.100 + 0.375i)23-s + (−3.55 + 3.51i)25-s + (−3.68 + 3.68i)27-s − 6.62i·29-s + (−0.897 + 0.518i)31-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.963i)3-s + (0.379 + 0.925i)5-s + (−0.940 − 0.340i)7-s + (0.00462 + 0.00266i)9-s + (0.333 + 0.578i)11-s + (0.530 + 0.530i)13-s + (−0.989 + 0.127i)15-s + (−1.01 − 0.272i)17-s + (−0.413 + 0.716i)19-s + (0.570 − 0.817i)21-s + (0.0209 + 0.0783i)23-s + (−0.711 + 0.702i)25-s + (−0.708 + 0.708i)27-s − 1.22i·29-s + (−0.161 + 0.0930i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.306475 + 1.03031i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.306475 + 1.03031i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.849 - 2.06i)T \) |
| 7 | \( 1 + (2.48 + 0.900i)T \) |
good | 3 | \( 1 + (0.447 - 1.66i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-1.10 - 1.91i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.91 - 1.91i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.19 + 1.12i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.80 - 3.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.100 - 0.375i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 6.62iT - 29T^{2} \) |
| 31 | \( 1 + (0.897 - 0.518i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.88 - 1.84i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 4.03iT - 41T^{2} \) |
| 43 | \( 1 + (-8.37 + 8.37i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.48 + 5.52i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (6.07 + 1.62i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.71 - 6.42i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.84 - 4.52i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.70 - 10.0i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 9.39T + 71T^{2} \) |
| 73 | \( 1 + (3.69 - 13.7i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.38 - 3.11i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.77 - 7.77i)T + 83iT^{2} \) |
| 89 | \( 1 + (5.20 - 9.01i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.1 + 12.1i)T - 97iT^{2} \) |
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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
−10.90903304467941540703555310939, −10.12958109839082797125979962253, −9.725619712258450866733646725859, −8.767325061643169310294765159635, −7.24100325506451977511915508984, −6.61785739279368143121477893434, −5.67291549818388319464795391440, −4.28231590281303695444122699244, −3.65069289413246266228219991863, −2.17817837927299267336124019232,
0.62508544887794534781469225650, 2.00868905753474267664707957913, 3.52946598296288453780496612442, 4.91303971387284914000187884748, 6.14431714030095213674608688388, 6.47384584470744053592627757811, 7.68938527937934352802020292094, 8.843977122010524349545435951522, 9.198571323464741219247491824599, 10.46390950187938679226980905790