| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (0.959 − 0.281i)6-s + (−2.73 − 1.76i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.841 + 0.540i)10-s + (0.635 + 4.42i)11-s + (0.142 + 0.989i)12-s + (2.77 − 1.78i)13-s + (2.13 − 2.46i)14-s + (0.415 − 0.909i)15-s + (0.841 + 0.540i)16-s + (1.84 − 0.540i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.292 + 0.337i)5-s + (0.391 − 0.115i)6-s + (−1.03 − 0.665i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.266 + 0.170i)10-s + (0.191 + 1.33i)11-s + (0.0410 + 0.285i)12-s + (0.769 − 0.494i)13-s + (0.569 − 0.657i)14-s + (0.107 − 0.234i)15-s + (0.210 + 0.135i)16-s + (0.446 − 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11974 + 0.481723i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11974 + 0.481723i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-2.83 - 3.86i)T \) |
| good | 7 | \( 1 + (2.73 + 1.76i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.635 - 4.42i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.77 + 1.78i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.84 + 0.540i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-3.60 - 1.05i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-7.45 + 2.19i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.72 - 3.78i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-6.24 + 7.21i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.324 - 0.373i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.18 - 2.60i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 8.80T + 47T^{2} \) |
| 53 | \( 1 + (3.03 + 1.95i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (1.37 - 0.882i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.11 - 6.83i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (2.11 - 14.6i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.834 - 5.80i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.37 - 0.992i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (11.1 - 7.17i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.57 + 6.43i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (4.54 + 9.94i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (3.42 + 3.95i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.24830775418434342948752968155, −9.830644518182210213823094239675, −8.837119043591866429519059613642, −7.51351060254718660375174754535, −7.18202182125380329312273910125, −6.26326145643642666940222788223, −5.47947331914399311410900826570, −4.16354428349532606179325430622, −2.96158316129061065516368602161, −1.13089216324243634805371720424,
0.902862045540364869988173161515, 2.79858075981336899032073162123, 3.53922354015277974245159106060, 4.78258399958693809889596034893, 5.87613688304875077305279689424, 6.45184344147625923062963962743, 8.147260588621568601202847964382, 9.032790692612080832908763100764, 9.389667379729291245894861195376, 10.41162524563719407646595445466
