| L(s) = 1 | − 2-s + 0.823·3-s + 4-s + 1.87·5-s − 0.823·6-s + 2.57·7-s − 8-s − 2.32·9-s − 1.87·10-s + 5.41·11-s + 0.823·12-s + 0.617·13-s − 2.57·14-s + 1.54·15-s + 16-s − 2.34·17-s + 2.32·18-s + 2.22·19-s + 1.87·20-s + 2.12·21-s − 5.41·22-s − 0.533·23-s − 0.823·24-s − 1.47·25-s − 0.617·26-s − 4.38·27-s + 2.57·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.475·3-s + 0.5·4-s + 0.840·5-s − 0.336·6-s + 0.974·7-s − 0.353·8-s − 0.774·9-s − 0.594·10-s + 1.63·11-s + 0.237·12-s + 0.171·13-s − 0.688·14-s + 0.399·15-s + 0.250·16-s − 0.567·17-s + 0.547·18-s + 0.510·19-s + 0.420·20-s + 0.463·21-s − 1.15·22-s − 0.111·23-s − 0.168·24-s − 0.294·25-s − 0.121·26-s − 0.843·27-s + 0.487·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.470710025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.470710025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 0.823T + 3T^{2} \) |
| 5 | \( 1 - 1.87T + 5T^{2} \) |
| 7 | \( 1 - 2.57T + 7T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 - 0.617T + 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 19 | \( 1 - 2.22T + 19T^{2} \) |
| 23 | \( 1 + 0.533T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 - 5.06T + 47T^{2} \) |
| 53 | \( 1 - 1.84T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 0.829T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 5.04T + 73T^{2} \) |
| 79 | \( 1 - 3.19T + 79T^{2} \) |
| 83 | \( 1 - 9.27T + 83T^{2} \) |
| 89 | \( 1 + 6.47T + 89T^{2} \) |
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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
−8.173013599500937315329790743072, −7.59086944308617261253897099680, −6.67189857064322489130587704235, −6.04805256897202183786115820030, −5.42169204772661046170167984952, −4.34743321957611400688274665488, −3.55330177341011382001584576364, −2.44513481729863965683262395041, −1.84278188549748264850235496600, −0.945042311473105300232065447150,
0.945042311473105300232065447150, 1.84278188549748264850235496600, 2.44513481729863965683262395041, 3.55330177341011382001584576364, 4.34743321957611400688274665488, 5.42169204772661046170167984952, 6.04805256897202183786115820030, 6.67189857064322489130587704235, 7.59086944308617261253897099680, 8.173013599500937315329790743072
