L(s) = 1 | + (0.647 + 1.12i)3-s + i·5-s + (1.06 + 0.613i)7-s + (0.660 − 1.14i)9-s + (2.37 − 1.37i)11-s + (0.155 − 3.60i)13-s + (−1.12 + 0.647i)15-s + (2.79 − 4.83i)17-s + (4.07 + 2.35i)19-s + 1.59i·21-s + (0.674 + 1.16i)23-s − 25-s + 5.59·27-s + (−0.168 − 0.292i)29-s + 2.92i·31-s + ⋯ |
L(s) = 1 | + (0.374 + 0.647i)3-s + 0.447i·5-s + (0.401 + 0.231i)7-s + (0.220 − 0.381i)9-s + (0.716 − 0.413i)11-s + (0.0431 − 0.999i)13-s + (−0.289 + 0.167i)15-s + (0.677 − 1.17i)17-s + (0.935 + 0.540i)19-s + 0.347i·21-s + (0.140 + 0.243i)23-s − 0.200·25-s + 1.07·27-s + (−0.0313 − 0.0543i)29-s + 0.526i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.116693181\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.116693181\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-0.155 + 3.60i)T \) |
good | 3 | \( 1 + (-0.647 - 1.12i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.06 - 0.613i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.37 + 1.37i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.79 + 4.83i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.07 - 2.35i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.674 - 1.16i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.168 + 0.292i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.92iT - 31T^{2} \) |
| 37 | \( 1 + (1.53 - 0.884i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.36 - 4.82i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.71 - 6.43i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.95iT - 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + (5.10 + 2.94i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.82 - 8.36i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.58 + 5.53i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.97 + 2.87i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 5.45iT - 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 1.01iT - 83T^{2} \) |
| 89 | \( 1 + (-9.60 + 5.54i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.75 - 2.16i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.900138597971014796974109812103, −9.314426732344896715274691877987, −8.396075323137252788111455415184, −7.57271711429644740320510557408, −6.63851600983579817868125428407, −5.59862675492087456254941864887, −4.75094381065492844658055674539, −3.46968462063143047423069545208, −3.03049418461065005052838936646, −1.20808483565098098752016492110,
1.30377955131180550174603021177, 2.08232664344585830615159807868, 3.65856502980039504652473003143, 4.56341969029600745041307679926, 5.53030988956018997117761899454, 6.78054548622656622545916910127, 7.29223465993202561265550346461, 8.255798231445372388863188873968, 8.855958380833381520963238686411, 9.810766368744374854194521000863