L(s) = 1 | + (0.439 − 0.761i)2-s + (1.11 + 1.32i)3-s + (0.613 + 1.06i)4-s − 1.34·5-s + (1.5 − 0.264i)6-s + 2.83·8-s + (−0.520 + 2.95i)9-s + (−0.592 + 1.02i)10-s + 1.65·11-s + (−0.726 + 1.99i)12-s + (−1.68 + 2.91i)13-s + (−1.5 − 1.78i)15-s + (0.0209 − 0.0362i)16-s + (0.233 − 0.405i)17-s + (2.02 + 1.69i)18-s + (−1.61 − 2.79i)19-s + ⋯ |
L(s) = 1 | + (0.310 − 0.538i)2-s + (0.642 + 0.766i)3-s + (0.306 + 0.531i)4-s − 0.602·5-s + (0.612 − 0.107i)6-s + 1.00·8-s + (−0.173 + 0.984i)9-s + (−0.187 + 0.324i)10-s + 0.498·11-s + (−0.209 + 0.576i)12-s + (−0.467 + 0.809i)13-s + (−0.387 − 0.461i)15-s + (0.00523 − 0.00906i)16-s + (0.0567 − 0.0982i)17-s + (0.476 + 0.399i)18-s + (−0.370 − 0.641i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.87184 + 0.784458i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87184 + 0.784458i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.11 - 1.32i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.439 + 0.761i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + (1.68 - 2.91i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.233 + 0.405i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.61 + 2.79i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + (3.13 + 5.42i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.61 - 7.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.61 + 7.99i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.70 + 2.95i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.20 - 3.82i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.67 + 8.10i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.286 + 0.497i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.19 + 9.00i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.81 + 6.61i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.298 + 0.516i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 0.554T + 71T^{2} \) |
| 73 | \( 1 + (-1.02 + 1.77i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.20 + 2.08i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.52 + 13.0i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.54 - 7.86i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.949 + 1.64i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.25671461459845713030920012265, −10.56379653417027652622457286155, −9.375678763818626664947178045532, −8.665732836084758500486423050286, −7.60554651637607125938360216187, −6.84997138915573757726807970200, −4.97936377199446691309229423952, −4.14746444960558267365807583154, −3.30322834398127462867770409850, −2.16362808756966934270816126251,
1.23454846392654247227327934858, 2.79722891893787142943577745980, 4.14332904254177473246108783603, 5.47414082987462139635001727830, 6.46619058874958595268693337597, 7.32461559070774217535956468217, 7.942787735238470584668901909057, 9.023700141116652858582322621536, 10.07479174096784668167605742988, 11.11230668527169531739935341145