| L(s) = 1 | + 2-s + 3.07·3-s + 4-s + 2.26·5-s + 3.07·6-s − 4.26·7-s + 8-s + 6.42·9-s + 2.26·10-s + 11-s + 3.07·12-s + 0.476·13-s − 4.26·14-s + 6.93·15-s + 16-s + 2.57·17-s + 6.42·18-s − 7.32·19-s + 2.26·20-s − 13.0·21-s + 22-s − 9.23·23-s + 3.07·24-s + 0.108·25-s + 0.476·26-s + 10.5·27-s − 4.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.77·3-s + 0.5·4-s + 1.01·5-s + 1.25·6-s − 1.61·7-s + 0.353·8-s + 2.14·9-s + 0.714·10-s + 0.301·11-s + 0.886·12-s + 0.132·13-s − 1.13·14-s + 1.79·15-s + 0.250·16-s + 0.623·17-s + 1.51·18-s − 1.68·19-s + 0.505·20-s − 2.85·21-s + 0.213·22-s − 1.92·23-s + 0.626·24-s + 0.0216·25-s + 0.0935·26-s + 2.02·27-s − 0.805·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.373369787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.373369787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 3.07T + 3T^{2} \) |
| 5 | \( 1 - 2.26T + 5T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 13 | \( 1 - 0.476T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 + 7.32T + 19T^{2} \) |
| 23 | \( 1 + 9.23T + 23T^{2} \) |
| 29 | \( 1 + 0.0534T + 29T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 - 5.06T + 37T^{2} \) |
| 41 | \( 1 + 0.892T + 41T^{2} \) |
| 47 | \( 1 + 3.37T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 1.34T + 61T^{2} \) |
| 67 | \( 1 - 8.40T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 6.90T + 73T^{2} \) |
| 79 | \( 1 + 0.381T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 8.66T + 97T^{2} \) |
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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
−9.959255138363590232412358557040, −9.309474871467123270654290901946, −8.455766067043033190088767083621, −7.54495479271107039068858311295, −6.37028519755003386136221656313, −6.07939333539354996335385737572, −4.32417739608057650913571110168, −3.58094897505641784596250556752, −2.68164140051741460442134557967, −1.91208257967013585391190612124,
1.91208257967013585391190612124, 2.68164140051741460442134557967, 3.58094897505641784596250556752, 4.32417739608057650913571110168, 6.07939333539354996335385737572, 6.37028519755003386136221656313, 7.54495479271107039068858311295, 8.455766067043033190088767083621, 9.309474871467123270654290901946, 9.959255138363590232412358557040
