| L(s) = 1 | + 2-s + 0.255·3-s + 4-s − 3.96·5-s + 0.255·6-s + 1.56·7-s + 8-s − 2.93·9-s − 3.96·10-s − 0.915·11-s + 0.255·12-s + 2.29·13-s + 1.56·14-s − 1.01·15-s + 16-s + 5.82·17-s − 2.93·18-s − 6.98·19-s − 3.96·20-s + 0.398·21-s − 0.915·22-s + 5.36·23-s + 0.255·24-s + 10.7·25-s + 2.29·26-s − 1.51·27-s + 1.56·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.147·3-s + 0.5·4-s − 1.77·5-s + 0.104·6-s + 0.589·7-s + 0.353·8-s − 0.978·9-s − 1.25·10-s − 0.276·11-s + 0.0737·12-s + 0.635·13-s + 0.416·14-s − 0.261·15-s + 0.250·16-s + 1.41·17-s − 0.691·18-s − 1.60·19-s − 0.886·20-s + 0.0870·21-s − 0.195·22-s + 1.11·23-s + 0.0521·24-s + 2.14·25-s + 0.449·26-s − 0.291·27-s + 0.294·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 - 0.255T + 3T^{2} \) |
| 5 | \( 1 + 3.96T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + 0.915T + 11T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 17 | \( 1 - 5.82T + 17T^{2} \) |
| 19 | \( 1 + 6.98T + 19T^{2} \) |
| 23 | \( 1 - 5.36T + 23T^{2} \) |
| 29 | \( 1 - 0.293T + 29T^{2} \) |
| 31 | \( 1 - 7.27T + 31T^{2} \) |
| 37 | \( 1 + 6.31T + 37T^{2} \) |
| 41 | \( 1 + 5.07T + 41T^{2} \) |
| 43 | \( 1 + 9.20T + 43T^{2} \) |
| 47 | \( 1 + 9.76T + 47T^{2} \) |
| 53 | \( 1 + 8.27T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 8.68T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 5.06T + 71T^{2} \) |
| 73 | \( 1 + 8.96T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 4.52T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 3.16T + 97T^{2} \) |
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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
−7.925194280503498263056599204889, −6.92866218989411972540003860414, −6.42923782164007877862088096154, −5.23936775353492706702871350297, −4.89996933847880214157547704967, −3.90734208300016454086240080680, −3.40544982515619071576715466546, −2.70771834267243654295216468442, −1.33451123418424038023127631575, 0,
1.33451123418424038023127631575, 2.70771834267243654295216468442, 3.40544982515619071576715466546, 3.90734208300016454086240080680, 4.89996933847880214157547704967, 5.23936775353492706702871350297, 6.42923782164007877862088096154, 6.92866218989411972540003860414, 7.925194280503498263056599204889
