| L(s) = 1 | + 0.554·2-s + 2.69·3-s − 1.69·4-s − 0.109·5-s + 1.49·6-s − 2.24·7-s − 2.04·8-s + 4.24·9-s − 0.0609·10-s − 1.19·11-s − 4.55·12-s − 1.69·13-s − 1.24·14-s − 0.295·15-s + 2.24·16-s + 4.46·17-s + 2.35·18-s + 5.85·19-s + 0.185·20-s − 6.04·21-s − 0.664·22-s − 1.66·23-s − 5.51·24-s − 4.98·25-s − 0.939·26-s + 3.35·27-s + 3.80·28-s + ⋯ |
| L(s) = 1 | + 0.392·2-s + 1.55·3-s − 0.846·4-s − 0.0491·5-s + 0.609·6-s − 0.849·7-s − 0.724·8-s + 1.41·9-s − 0.0192·10-s − 0.361·11-s − 1.31·12-s − 0.469·13-s − 0.333·14-s − 0.0764·15-s + 0.561·16-s + 1.08·17-s + 0.555·18-s + 1.34·19-s + 0.0415·20-s − 1.31·21-s − 0.141·22-s − 0.347·23-s − 1.12·24-s − 0.997·25-s − 0.184·26-s + 0.646·27-s + 0.718·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 - 0.554T + 2T^{2} \) |
| 3 | \( 1 - 2.69T + 3T^{2} \) |
| 5 | \( 1 + 0.109T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 - 0.692T + 29T^{2} \) |
| 31 | \( 1 + 8.23T + 31T^{2} \) |
| 37 | \( 1 - 0.951T + 37T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 1.97T + 47T^{2} \) |
| 53 | \( 1 - 9.15T + 53T^{2} \) |
| 59 | \( 1 + 3.95T + 59T^{2} \) |
| 61 | \( 1 + 3.97T + 61T^{2} \) |
| 67 | \( 1 + 9.03T + 67T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 3.56T + 79T^{2} \) |
| 83 | \( 1 + 9.04T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 1.25T + 97T^{2} \) |
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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
−7.62706796309578128885259658901, −7.10537739759863471596582328269, −5.81088876997351093811127113812, −5.47422755172514686710534748958, −4.39240038018184669266111952357, −3.70903960877792966362064647178, −3.20170345556285803121880127906, −2.63175337795686567699418164084, −1.43219474102881439221180514668, 0,
1.43219474102881439221180514668, 2.63175337795686567699418164084, 3.20170345556285803121880127906, 3.70903960877792966362064647178, 4.39240038018184669266111952357, 5.47422755172514686710534748958, 5.81088876997351093811127113812, 7.10537739759863471596582328269, 7.62706796309578128885259658901
