L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.499i)4-s + (−0.781 − 0.781i)5-s + (−0.965 + 0.258i)6-s + (−1.20 + 2.35i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (0.552 + 0.957i)10-s + (0.312 − 1.16i)11-s + 12-s + (2.62 − 2.47i)13-s + (1.77 − 1.96i)14-s + (−1.06 − 0.286i)15-s + (0.500 + 0.866i)16-s + (1.19 − 2.06i)17-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 + 0.249i)4-s + (−0.349 − 0.349i)5-s + (−0.394 + 0.105i)6-s + (−0.455 + 0.890i)7-s + (−0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (0.174 + 0.302i)10-s + (0.0943 − 0.352i)11-s + 0.288·12-s + (0.728 − 0.685i)13-s + (0.474 − 0.524i)14-s + (−0.275 − 0.0739i)15-s + (0.125 + 0.216i)16-s + (0.289 − 0.501i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.997089 - 0.579286i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.997089 - 0.579286i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (1.20 - 2.35i)T \) |
| 13 | \( 1 + (-2.62 + 2.47i)T \) |
good | 5 | \( 1 + (0.781 + 0.781i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.312 + 1.16i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.19 + 2.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.87 + 1.30i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-5.04 + 2.91i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.18 + 7.25i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.07 - 6.07i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.63 - 9.84i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.0706 + 0.263i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (9.39 + 5.42i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.79 + 4.79i)T - 47iT^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 + (-2.46 - 9.18i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-9.83 - 5.67i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.76 + 1.54i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (2.24 + 8.36i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-4.16 + 4.16i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.38T + 79T^{2} \) |
| 83 | \( 1 + (2.18 + 2.18i)T + 83iT^{2} \) |
| 89 | \( 1 + (15.1 + 4.06i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (1.77 - 0.476i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.45987590203301399263388397925, −9.661487139851545419032798830281, −8.642867730547323211126653400712, −8.388536613089310075835724870581, −7.22356797520281720297229161107, −6.25638924758564074789762533072, −5.10897525615272749029011626271, −3.45390195856179017977450573509, −2.63795719392355749902467091398, −0.908092114782060731933739743721,
1.42142317384242860888758785788, 3.20532049838466860509864606773, 3.98892370930322398747015491191, 5.47853883598342973479793706198, 6.83774006526858761268593234124, 7.34406947838835145907149890870, 8.279705259838595649748999654668, 9.359990675596747639055249085290, 9.820307063576630829528956573637, 10.91709897156650687690928764696