| L(s) = 1 | + 0.735·2-s + 0.544·3-s − 1.45·4-s + 0.962·5-s + 0.400·6-s − 3.25·7-s − 2.54·8-s − 2.70·9-s + 0.707·10-s − 0.883·11-s − 0.794·12-s − 4.48·13-s − 2.39·14-s + 0.523·15-s + 1.04·16-s − 3.18·17-s − 1.98·18-s + 4.12·19-s − 1.40·20-s − 1.77·21-s − 0.649·22-s + 3.73·23-s − 1.38·24-s − 4.07·25-s − 3.29·26-s − 3.10·27-s + 4.74·28-s + ⋯ |
| L(s) = 1 | + 0.519·2-s + 0.314·3-s − 0.729·4-s + 0.430·5-s + 0.163·6-s − 1.22·7-s − 0.899·8-s − 0.901·9-s + 0.223·10-s − 0.266·11-s − 0.229·12-s − 1.24·13-s − 0.639·14-s + 0.135·15-s + 0.262·16-s − 0.771·17-s − 0.468·18-s + 0.946·19-s − 0.314·20-s − 0.386·21-s − 0.138·22-s + 0.779·23-s − 0.282·24-s − 0.814·25-s − 0.646·26-s − 0.597·27-s + 0.897·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 + T \) |
| good | 2 | \( 1 - 0.735T + 2T^{2} \) |
| 3 | \( 1 - 0.544T + 3T^{2} \) |
| 5 | \( 1 - 0.962T + 5T^{2} \) |
| 7 | \( 1 + 3.25T + 7T^{2} \) |
| 11 | \( 1 + 0.883T + 11T^{2} \) |
| 13 | \( 1 + 4.48T + 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 23 | \( 1 - 3.73T + 23T^{2} \) |
| 29 | \( 1 - 6.33T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 + 9.31T + 43T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 + 7.06T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 + 9.15T + 67T^{2} \) |
| 71 | \( 1 + 4.57T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 7.20T + 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 - 0.838T + 89T^{2} \) |
| 97 | \( 1 - 0.475T + 97T^{2} \) |
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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
−10.00828728792462251160152786014, −9.509859634862580768198524283937, −8.828391170310390024604574655929, −7.68106061500103327823277636609, −6.48784395151949075309761480761, −5.61363761233904359929176108641, −4.73436831617832172151711616752, −3.37748843144656817619587842063, −2.63646459844130186224277379099, 0,
2.63646459844130186224277379099, 3.37748843144656817619587842063, 4.73436831617832172151711616752, 5.61363761233904359929176108641, 6.48784395151949075309761480761, 7.68106061500103327823277636609, 8.828391170310390024604574655929, 9.509859634862580768198524283937, 10.00828728792462251160152786014
