L(s) = 1 | + (2.15 − 1.56i)2-s + (−0.234 + 0.721i)3-s + (1.57 − 4.85i)4-s + (0.625 + 1.92i)6-s + 2.04·7-s + (−2.55 − 7.86i)8-s + (1.96 + 1.42i)9-s + (−1.09 + 0.792i)11-s + (3.13 + 2.27i)12-s + (1.06 + 0.775i)13-s + (4.40 − 3.19i)14-s + (−9.56 − 6.95i)16-s + (−1.26 − 3.88i)17-s + 6.45·18-s + (−1.51 − 4.64i)19-s + ⋯ |
L(s) = 1 | + (1.52 − 1.10i)2-s + (−0.135 + 0.416i)3-s + (0.788 − 2.42i)4-s + (0.255 + 0.785i)6-s + 0.771·7-s + (−0.903 − 2.77i)8-s + (0.653 + 0.474i)9-s + (−0.328 + 0.239i)11-s + (0.904 + 0.657i)12-s + (0.296 + 0.215i)13-s + (1.17 − 0.854i)14-s + (−2.39 − 1.73i)16-s + (−0.306 − 0.942i)17-s + 1.52·18-s + (−0.346 − 1.06i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0209 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0209 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.48558 - 2.43406i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.48558 - 2.43406i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-2.15 + 1.56i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.234 - 0.721i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 + (1.09 - 0.792i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.06 - 0.775i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.26 + 3.88i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.51 + 4.64i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.21 - 1.60i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.42 - 4.39i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.21 - 6.80i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (6.99 + 5.08i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-8.17 - 5.93i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.43T + 43T^{2} \) |
| 47 | \( 1 + (2.33 - 7.20i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.191 + 0.590i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-9.17 - 6.66i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.523 - 0.380i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (3.38 + 10.4i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (0.912 - 2.80i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-10.9 + 7.95i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.583 + 1.79i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.733 + 2.25i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (5.93 - 4.31i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.79 - 5.51i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.86593773328441789768360296556, −9.962266856869548710988327874596, −9.032748174587496906220115536209, −7.48442787239382438666409587357, −6.47712227055950762662375000714, −5.08065595653221829616475945579, −4.88236608580364639540783937581, −3.87085978297438778146723780881, −2.62845431821111200384828755758, −1.52574057981633482419162874480,
2.06956332681348278142918572623, 3.71089900643785780105089010996, 4.32608366278836485565716455678, 5.53152196277037094229571415684, 6.17442010396192962000876472405, 7.00565949109662279101179495412, 8.009413606734150913463182285140, 8.375717060431021513261974818106, 10.06802786007193265956909065063, 11.24376260980153875773037567834