L(s) = 1 | + (0.991 − 0.720i)2-s + (−0.153 + 0.472i)4-s + (0.809 + 0.587i)5-s + (−0.139 + 0.429i)7-s + (0.945 + 2.91i)8-s + 1.22·10-s + (−1.55 + 2.93i)11-s + (3.91 − 2.84i)13-s + (0.171 + 0.526i)14-s + (2.23 + 1.62i)16-s + (−0.598 − 0.435i)17-s + (2.10 + 6.47i)19-s + (−0.402 + 0.292i)20-s + (0.572 + 4.02i)22-s − 0.00634·23-s + ⋯ |
L(s) = 1 | + (0.701 − 0.509i)2-s + (−0.0767 + 0.236i)4-s + (0.361 + 0.262i)5-s + (−0.0527 + 0.162i)7-s + (0.334 + 1.02i)8-s + 0.387·10-s + (−0.468 + 0.883i)11-s + (1.08 − 0.789i)13-s + (0.0457 + 0.140i)14-s + (0.558 + 0.405i)16-s + (−0.145 − 0.105i)17-s + (0.482 + 1.48i)19-s + (−0.0898 + 0.0653i)20-s + (0.122 + 0.858i)22-s − 0.00132·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.03279 + 0.445641i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03279 + 0.445641i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (1.55 - 2.93i)T \) |
good | 2 | \( 1 + (-0.991 + 0.720i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (0.139 - 0.429i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.91 + 2.84i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.598 + 0.435i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.10 - 6.47i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 0.00634T + 23T^{2} \) |
| 29 | \( 1 + (0.100 - 0.308i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.53 + 3.29i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.27 + 7.00i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (3.39 + 10.4i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 1.80T + 43T^{2} \) |
| 47 | \( 1 + (0.518 + 1.59i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (6.98 - 5.07i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.463 - 1.42i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (10.5 + 7.66i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 9.60T + 67T^{2} \) |
| 71 | \( 1 + (9.23 + 6.70i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.16 + 9.73i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.69 + 1.23i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-12.8 - 9.34i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 + (4.75 - 3.45i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.98952822711531797193700976835, −10.40346230731241794279804812701, −9.337755632481345643066838698483, −8.197202821677847827359712631968, −7.51470363447875901178755418798, −6.07740202165578278922821569396, −5.27759704485057092140121761286, −4.07785170019308857095877258743, −3.12372392842868037598141737382, −1.93343746196572892769213435764,
1.15161825299463361527328229204, 3.10300840809347635827001295514, 4.39854465889474649593397153173, 5.21555702062260132464667682434, 6.24716761443637348673965312481, 6.80321971271261960133808014321, 8.200054508244830469330520020151, 9.083528768108784844569077408150, 9.974839429337604130172217814775, 10.94058076391149749264300738791