| L(s) = 1 | + 2-s − 1.95·3-s + 4-s + 0.613·5-s − 1.95·6-s + 0.465·7-s + 8-s + 0.829·9-s + 0.613·10-s + 4.22·11-s − 1.95·12-s + 0.465·14-s − 1.20·15-s + 16-s + 5.33·17-s + 0.829·18-s + 19-s + 0.613·20-s − 0.911·21-s + 4.22·22-s − 6.57·23-s − 1.95·24-s − 4.62·25-s + 4.24·27-s + 0.465·28-s + 8.11·29-s − 1.20·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.12·3-s + 0.5·4-s + 0.274·5-s − 0.798·6-s + 0.176·7-s + 0.353·8-s + 0.276·9-s + 0.194·10-s + 1.27·11-s − 0.564·12-s + 0.124·14-s − 0.310·15-s + 0.250·16-s + 1.29·17-s + 0.195·18-s + 0.229·19-s + 0.137·20-s − 0.198·21-s + 0.900·22-s − 1.37·23-s − 0.399·24-s − 0.924·25-s + 0.817·27-s + 0.0880·28-s + 1.50·29-s − 0.219·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.546316864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.546316864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 - 0.613T + 5T^{2} \) |
| 7 | \( 1 - 0.465T + 7T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 17 | \( 1 - 5.33T + 17T^{2} \) |
| 23 | \( 1 + 6.57T + 23T^{2} \) |
| 29 | \( 1 - 8.11T + 29T^{2} \) |
| 31 | \( 1 + 4.04T + 31T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 - 4.71T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 3.74T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 67 | \( 1 - 9.47T + 67T^{2} \) |
| 71 | \( 1 + 16.6T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 6.66T + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.80882643821126660217451807125, −7.10972860400845745389917807451, −6.23585246284264180965681796031, −5.92200906555318991211607893791, −5.35496024005537663109621145084, −4.42286780513621372193415169930, −3.87232120342845961829811076677, −2.87343376954607028625081004971, −1.73125918287641820139170665768, −0.826193564310661629607336569353,
0.826193564310661629607336569353, 1.73125918287641820139170665768, 2.87343376954607028625081004971, 3.87232120342845961829811076677, 4.42286780513621372193415169930, 5.35496024005537663109621145084, 5.92200906555318991211607893791, 6.23585246284264180965681796031, 7.10972860400845745389917807451, 7.80882643821126660217451807125
