| L(s) = 1 | + 2-s + 0.0419·3-s + 4-s − 2.69·5-s + 0.0419·6-s − 1.02·7-s + 8-s − 2.99·9-s − 2.69·10-s + 2.48·11-s + 0.0419·12-s + 0.490·13-s − 1.02·14-s − 0.112·15-s + 16-s + 17-s − 2.99·18-s + 6.62·19-s − 2.69·20-s − 0.0430·21-s + 2.48·22-s − 5.34·23-s + 0.0419·24-s + 2.25·25-s + 0.490·26-s − 0.251·27-s − 1.02·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0242·3-s + 0.5·4-s − 1.20·5-s + 0.0171·6-s − 0.388·7-s + 0.353·8-s − 0.999·9-s − 0.851·10-s + 0.750·11-s + 0.0121·12-s + 0.136·13-s − 0.274·14-s − 0.0291·15-s + 0.250·16-s + 0.242·17-s − 0.706·18-s + 1.52·19-s − 0.602·20-s − 0.00939·21-s + 0.530·22-s − 1.11·23-s + 0.00855·24-s + 0.451·25-s + 0.0961·26-s − 0.0484·27-s − 0.194·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.052747090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.052747090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0419T + 3T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 13 | \( 1 - 0.490T + 13T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 - 8.67T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 - 3.94T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 - 9.01T + 43T^{2} \) |
| 47 | \( 1 + 8.43T + 47T^{2} \) |
| 53 | \( 1 + 3.97T + 53T^{2} \) |
| 61 | \( 1 + 2.69T + 61T^{2} \) |
| 67 | \( 1 - 4.86T + 67T^{2} \) |
| 71 | \( 1 - 4.02T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 5.83T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 7.72T + 97T^{2} \) |
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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
−9.166088888445192282863322859243, −7.984072080794510053063288662569, −7.82757360596916538988687451069, −6.59492738733048862783954791463, −6.04921976480531857572171125534, −5.02758766774193189235907227770, −4.11140235377233215832568829220, −3.42033607890755679047348947531, −2.63615087854550200757547393889, −0.863524581637105268233993962547,
0.863524581637105268233993962547, 2.63615087854550200757547393889, 3.42033607890755679047348947531, 4.11140235377233215832568829220, 5.02758766774193189235907227770, 6.04921976480531857572171125534, 6.59492738733048862783954791463, 7.82757360596916538988687451069, 7.984072080794510053063288662569, 9.166088888445192282863322859243
