L(s) = 1 | + (−0.677 + 1.59i)3-s − 2.66·5-s + (0.654 − 2.56i)7-s + (−2.08 − 2.15i)9-s + 3.98·11-s + (1.00 − 1.73i)13-s + (1.80 − 4.25i)15-s + (−3.57 + 6.18i)17-s + (4.01 + 6.96i)19-s + (3.64 + 2.78i)21-s + 0.887·23-s + 2.12·25-s + (4.85 − 1.85i)27-s + (−1.35 − 2.33i)29-s + (−0.614 − 1.06i)31-s + ⋯ |
L(s) = 1 | + (−0.391 + 0.920i)3-s − 1.19·5-s + (0.247 − 0.968i)7-s + (−0.694 − 0.719i)9-s + 1.20·11-s + (0.277 − 0.480i)13-s + (0.466 − 1.09i)15-s + (−0.866 + 1.50i)17-s + (0.922 + 1.59i)19-s + (0.794 + 0.606i)21-s + 0.185·23-s + 0.424·25-s + (0.934 − 0.357i)27-s + (−0.250 − 0.434i)29-s + (−0.110 − 0.191i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9561150946\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9561150946\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.677 - 1.59i)T \) |
| 7 | \( 1 + (-0.654 + 2.56i)T \) |
good | 5 | \( 1 + 2.66T + 5T^{2} \) |
| 11 | \( 1 - 3.98T + 11T^{2} \) |
| 13 | \( 1 + (-1.00 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.57 - 6.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.01 - 6.96i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 0.887T + 23T^{2} \) |
| 29 | \( 1 + (1.35 + 2.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.614 + 1.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.26 - 9.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.43 - 2.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.40 + 5.88i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.06 - 10.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.38 - 4.13i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.79 - 8.29i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.74 + 8.22i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.49 - 9.51i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.62T + 71T^{2} \) |
| 73 | \( 1 + (-2.01 + 3.48i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.514 - 0.891i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.26 - 9.12i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.72 - 2.99i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.12 + 1.94i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.24116011539316807257451031966, −9.542618694036268775828644975926, −8.347039281448678671729722852428, −7.936645740085381876767254490933, −6.71519655838300231502186106116, −5.93915730970407263547666819607, −4.63844371547346423830124931019, −3.87450613238083533465690767291, −3.54050367493735002905521120429, −1.18988624691971977026547347648,
0.54739689199055663094320575780, 2.10170112294155979792050652075, 3.27883912382147072968832129017, 4.60286062355542388191512703151, 5.37251768196607947859162197459, 6.70426672256814912005360240090, 7.02069045280937344433441147256, 8.008924107741526992521972621748, 8.917980131332099823534126898897, 9.359804615880651650245329599167