| L(s) = 1 | − 2-s − 0.949·3-s + 4-s − 3.26·5-s + 0.949·6-s + 3.22·7-s − 8-s − 2.09·9-s + 3.26·10-s − 5.15·11-s − 0.949·12-s + 3.20·13-s − 3.22·14-s + 3.09·15-s + 16-s + 4.62·17-s + 2.09·18-s − 19-s − 3.26·20-s − 3.06·21-s + 5.15·22-s − 3.01·23-s + 0.949·24-s + 5.66·25-s − 3.20·26-s + 4.83·27-s + 3.22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.547·3-s + 0.5·4-s − 1.46·5-s + 0.387·6-s + 1.21·7-s − 0.353·8-s − 0.699·9-s + 1.03·10-s − 1.55·11-s − 0.273·12-s + 0.888·13-s − 0.862·14-s + 0.800·15-s + 0.250·16-s + 1.12·17-s + 0.494·18-s − 0.229·19-s − 0.730·20-s − 0.668·21-s + 1.09·22-s − 0.629·23-s + 0.193·24-s + 1.13·25-s − 0.628·26-s + 0.931·27-s + 0.609·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5937817827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5937817827\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 0.949T + 3T^{2} \) |
| 5 | \( 1 + 3.26T + 5T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 + 5.15T + 11T^{2} \) |
| 13 | \( 1 - 3.20T + 13T^{2} \) |
| 17 | \( 1 - 4.62T + 17T^{2} \) |
| 23 | \( 1 + 3.01T + 23T^{2} \) |
| 29 | \( 1 - 8.93T + 29T^{2} \) |
| 31 | \( 1 - 4.37T + 31T^{2} \) |
| 37 | \( 1 + 3.61T + 37T^{2} \) |
| 41 | \( 1 + 8.25T + 41T^{2} \) |
| 43 | \( 1 - 5.82T + 43T^{2} \) |
| 47 | \( 1 + 1.00T + 47T^{2} \) |
| 53 | \( 1 + 5.20T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 0.600T + 61T^{2} \) |
| 67 | \( 1 + 5.63T + 67T^{2} \) |
| 71 | \( 1 - 1.87T + 71T^{2} \) |
| 73 | \( 1 + 7.71T + 73T^{2} \) |
| 79 | \( 1 + 3.33T + 79T^{2} \) |
| 83 | \( 1 + 4.70T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 4.01T + 97T^{2} \) |
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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
−7.967705792209565232827932569856, −7.51947962096180497589512332586, −6.51779766312880992754597527264, −5.75354761149853522572624470502, −5.02379479932188727075264255505, −4.45396786910516385960172648424, −3.35624120009051735413213480726, −2.72983985530402193381182197572, −1.44444233182837425949821399361, −0.46029446023006774602734410473,
0.46029446023006774602734410473, 1.44444233182837425949821399361, 2.72983985530402193381182197572, 3.35624120009051735413213480726, 4.45396786910516385960172648424, 5.02379479932188727075264255505, 5.75354761149853522572624470502, 6.51779766312880992754597527264, 7.51947962096180497589512332586, 7.967705792209565232827932569856
