L(s) = 1 | + (2.38 + 2.74i)5-s + (−3.67 − 2.36i)7-s + (0.389 + 2.70i)11-s + (−5.17 + 3.32i)13-s + (−1.26 + 0.370i)17-s + (5.16 + 1.51i)19-s + (−0.596 + 4.75i)23-s + (−1.17 + 8.14i)25-s + (−6.27 + 1.84i)29-s + (1.69 − 3.72i)31-s + (−2.26 − 15.7i)35-s + (−3.77 + 4.35i)37-s + (5.09 + 5.87i)41-s + (−3.90 − 8.56i)43-s + 5.07·47-s + ⋯ |
L(s) = 1 | + (1.06 + 1.22i)5-s + (−1.38 − 0.892i)7-s + (0.117 + 0.816i)11-s + (−1.43 + 0.922i)13-s + (−0.306 + 0.0898i)17-s + (1.18 + 0.348i)19-s + (−0.124 + 0.992i)23-s + (−0.234 + 1.62i)25-s + (−1.16 + 0.341i)29-s + (0.305 − 0.668i)31-s + (−0.382 − 2.65i)35-s + (−0.620 + 0.716i)37-s + (0.795 + 0.917i)41-s + (−0.596 − 1.30i)43-s + 0.740·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.580495 + 0.960205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.580495 + 0.960205i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (0.596 - 4.75i)T \) |
good | 5 | \( 1 + (-2.38 - 2.74i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (3.67 + 2.36i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.389 - 2.70i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (5.17 - 3.32i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.26 - 0.370i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-5.16 - 1.51i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (6.27 - 1.84i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.69 + 3.72i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (3.77 - 4.35i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-5.09 - 5.87i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.90 + 8.56i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 5.07T + 47T^{2} \) |
| 53 | \( 1 + (3.43 + 2.20i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.90 + 1.86i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (1.10 - 2.41i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.627 + 4.36i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.45 - 10.1i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.23 - 1.83i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (2.84 - 1.82i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (4.71 - 5.43i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.07 - 8.92i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (1.60 + 1.85i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.11536249439404668676149500286, −9.775545581865955513165017541344, −9.377490217561764759485246681732, −7.40102723974424771792739029110, −7.09903572666976808837292641227, −6.37347899086939399525106479062, −5.33839700336335683462407218855, −3.95999199255804631920645870171, −2.98220108341934835136729510183, −1.92801504457033699107654550553,
0.51420838811021627803095037409, 2.29734055346497104043029305388, 3.20133631184883582277424535598, 4.83994032445354376542867190979, 5.59600429827712807853879135989, 6.14207520042251826077585027267, 7.34884082961067048164553229473, 8.559022478014071877546277923946, 9.272536991979274437535968313289, 9.645536247278961379366685114633