| L(s) = 1 | − 2-s − 1.10·3-s + 4-s + 2.36·5-s + 1.10·6-s + 0.505·7-s − 8-s − 1.76·9-s − 2.36·10-s − 3.96·11-s − 1.10·12-s + 2.83·13-s − 0.505·14-s − 2.62·15-s + 16-s − 4.17·17-s + 1.76·18-s + 2.34·19-s + 2.36·20-s − 0.561·21-s + 3.96·22-s + 0.120·23-s + 1.10·24-s + 0.607·25-s − 2.83·26-s + 5.29·27-s + 0.505·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.640·3-s + 0.5·4-s + 1.05·5-s + 0.452·6-s + 0.191·7-s − 0.353·8-s − 0.589·9-s − 0.748·10-s − 1.19·11-s − 0.320·12-s + 0.786·13-s − 0.135·14-s − 0.678·15-s + 0.250·16-s − 1.01·17-s + 0.416·18-s + 0.538·19-s + 0.529·20-s − 0.122·21-s + 0.844·22-s + 0.0251·23-s + 0.226·24-s + 0.121·25-s − 0.555·26-s + 1.01·27-s + 0.0956·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 1.10T + 3T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 - 0.505T + 7T^{2} \) |
| 11 | \( 1 + 3.96T + 11T^{2} \) |
| 13 | \( 1 - 2.83T + 13T^{2} \) |
| 17 | \( 1 + 4.17T + 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 23 | \( 1 - 0.120T + 23T^{2} \) |
| 29 | \( 1 - 5.49T + 29T^{2} \) |
| 31 | \( 1 + 2.86T + 31T^{2} \) |
| 37 | \( 1 + 0.733T + 37T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 - 5.41T + 43T^{2} \) |
| 47 | \( 1 - 4.21T + 47T^{2} \) |
| 53 | \( 1 + 4.78T + 53T^{2} \) |
| 59 | \( 1 - 9.35T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 + 5.68T + 71T^{2} \) |
| 73 | \( 1 + 8.62T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 5.14T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.216325012066451606013131405632, −7.37879091920037286781029947821, −6.48515737453050416384924301319, −5.88776386720353490439710971627, −5.39144304382793956931737368349, −4.48631105916080263612806791062, −3.04290879918706087428399733907, −2.35046783255518040431190275519, −1.29741511725715938076851741639, 0,
1.29741511725715938076851741639, 2.35046783255518040431190275519, 3.04290879918706087428399733907, 4.48631105916080263612806791062, 5.39144304382793956931737368349, 5.88776386720353490439710971627, 6.48515737453050416384924301319, 7.37879091920037286781029947821, 8.216325012066451606013131405632
