| L(s) = 1 | − 2-s + 0.699·3-s + 4-s + 1.73·5-s − 0.699·6-s − 1.23·7-s − 8-s − 2.51·9-s − 1.73·10-s − 0.713·11-s + 0.699·12-s + 1.40·13-s + 1.23·14-s + 1.21·15-s + 16-s − 3.91·17-s + 2.51·18-s + 1.37·19-s + 1.73·20-s − 0.862·21-s + 0.713·22-s − 0.0900·23-s − 0.699·24-s − 1.98·25-s − 1.40·26-s − 3.85·27-s − 1.23·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.403·3-s + 0.5·4-s + 0.776·5-s − 0.285·6-s − 0.466·7-s − 0.353·8-s − 0.836·9-s − 0.549·10-s − 0.215·11-s + 0.201·12-s + 0.388·13-s + 0.329·14-s + 0.313·15-s + 0.250·16-s − 0.949·17-s + 0.591·18-s + 0.314·19-s + 0.388·20-s − 0.188·21-s + 0.152·22-s − 0.0187·23-s − 0.142·24-s − 0.396·25-s − 0.275·26-s − 0.741·27-s − 0.233·28-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 | \( 1 - 0.699T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 + 0.713T + 11T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 + 0.0900T + 23T^{2} \) |
| 29 | \( 1 - 0.740T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 2.98T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 + 6.10T + 43T^{2} \) |
| 47 | \( 1 + 3.37T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 4.03T + 59T^{2} \) |
| 61 | \( 1 + 3.61T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 6.54T + 71T^{2} \) |
| 73 | \( 1 + 9.87T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 + 8.61T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 - 5.81T + 97T^{2} \) |
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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
−8.313766762329877489722434168883, −7.47743054470158429181721082864, −6.53869186733970999497668997257, −6.08177974485853120105102745308, −5.28871652994139513611819552342, −4.16167877635416200500009429713, −3.01629628052128344059039811091, −2.50592526761904993344509305184, −1.44986094075778945354878066991, 0,
1.44986094075778945354878066991, 2.50592526761904993344509305184, 3.01629628052128344059039811091, 4.16167877635416200500009429713, 5.28871652994139513611819552342, 6.08177974485853120105102745308, 6.53869186733970999497668997257, 7.47743054470158429181721082864, 8.313766762329877489722434168883
