| L(s) = 1 | + 2-s + 0.821·3-s + 4-s + 3.37·5-s + 0.821·6-s − 5.13·7-s + 8-s − 2.32·9-s + 3.37·10-s − 1.78·11-s + 0.821·12-s + 4.39·13-s − 5.13·14-s + 2.76·15-s + 16-s + 6.07·17-s − 2.32·18-s − 19-s + 3.37·20-s − 4.21·21-s − 1.78·22-s − 4.63·23-s + 0.821·24-s + 6.35·25-s + 4.39·26-s − 4.37·27-s − 5.13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.474·3-s + 0.5·4-s + 1.50·5-s + 0.335·6-s − 1.94·7-s + 0.353·8-s − 0.775·9-s + 1.06·10-s − 0.537·11-s + 0.237·12-s + 1.22·13-s − 1.37·14-s + 0.714·15-s + 0.250·16-s + 1.47·17-s − 0.548·18-s − 0.229·19-s + 0.753·20-s − 0.920·21-s − 0.379·22-s − 0.966·23-s + 0.167·24-s + 1.27·25-s + 0.862·26-s − 0.841·27-s − 0.970·28-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
| \(L(1)\) |
\(\approx\) |
\(4.198685423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.198685423\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.821T + 3T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 + 5.13T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 - 4.39T + 13T^{2} \) |
| 17 | \( 1 - 6.07T + 17T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 + 6.47T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 6.64T + 43T^{2} \) |
| 47 | \( 1 - 2.22T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 - 8.88T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 0.602T + 67T^{2} \) |
| 71 | \( 1 - 9.73T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 1.18T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 + 9.24T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.84125282851199917871777953029, −6.79838000230911144067795700113, −6.18923294027308170647139533575, −5.84616513093960988100536911672, −5.40179361998222920246692251609, −4.06632611904578302107805536367, −3.31079956103105761247406805850, −2.83052927445341251085633273647, −2.16405832248826948370427081571, −0.877426701942868224593131785966,
0.877426701942868224593131785966, 2.16405832248826948370427081571, 2.83052927445341251085633273647, 3.31079956103105761247406805850, 4.06632611904578302107805536367, 5.40179361998222920246692251609, 5.84616513093960988100536911672, 6.18923294027308170647139533575, 6.79838000230911144067795700113, 7.84125282851199917871777953029
