| L(s) = 1 | + (0.646 − 1.25i)2-s + (0.913 − 1.47i)3-s + (−1.16 − 1.62i)4-s + (1.77 − 1.67i)5-s + (−1.26 − 2.09i)6-s + (−0.443 + 0.0518i)7-s + (−2.79 + 0.414i)8-s + (−1.33 − 2.68i)9-s + (−0.957 − 3.30i)10-s + (1.21 − 0.364i)11-s + (−3.45 + 0.229i)12-s + (1.07 − 0.0626i)13-s + (−0.221 + 0.591i)14-s + (−0.841 − 4.13i)15-s + (−1.28 + 3.78i)16-s + (−1.19 − 0.436i)17-s + ⋯ |
| L(s) = 1 | + (0.457 − 0.889i)2-s + (0.527 − 0.849i)3-s + (−0.582 − 0.812i)4-s + (0.791 − 0.747i)5-s + (−0.514 − 0.857i)6-s + (−0.167 + 0.0196i)7-s + (−0.989 + 0.146i)8-s + (−0.444 − 0.895i)9-s + (−0.302 − 1.04i)10-s + (0.367 − 0.109i)11-s + (−0.997 + 0.0661i)12-s + (0.298 − 0.0173i)13-s + (−0.0592 + 0.158i)14-s + (−0.217 − 1.06i)15-s + (−0.321 + 0.946i)16-s + (−0.290 − 0.105i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.327854 - 2.23345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.327854 - 2.23345i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.646 + 1.25i)T \) |
| 3 | \( 1 + (-0.913 + 1.47i)T \) |
| good | 5 | \( 1 + (-1.77 + 1.67i)T + (0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (0.443 - 0.0518i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (-1.21 + 0.364i)T + (9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (-1.07 + 0.0626i)T + (12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (1.19 + 0.436i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-2.47 - 6.80i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.611 - 0.0715i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (1.71 + 3.42i)T + (-17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (-4.22 + 5.66i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-0.264 - 0.315i)T + (-6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-3.83 + 2.52i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (0.716 - 3.02i)T + (-38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-5.65 - 7.59i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (2.41 - 1.39i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-11.0 - 3.31i)T + (49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (7.31 + 3.15i)T + (41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (3.18 - 6.34i)T + (-40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (2.04 + 11.6i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.0314 + 0.178i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (5.23 + 3.44i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-0.156 + 0.237i)T + (-32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (-0.476 + 2.70i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-4.61 + 4.88i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−9.985202782780402082021949766619, −9.405380784471659981904721786728, −8.668549869141578362352663749810, −7.66478037339852984409372466452, −6.16005154562616469218166280491, −5.78019119133466742361747518229, −4.37054721960148085637963411987, −3.25011283463057244486401664478, −2.03180116647657518173018415955, −1.08768594411183189696090103944,
2.56717813912343012724708613532, 3.44798325783745402724512384413, 4.59879743118934431527408811149, 5.46639632224661378780118625278, 6.53336054169523471958678357359, 7.20426525783990276596885830275, 8.459177130056999681399493344878, 9.131521253788921516084654053175, 9.877701482938211846989495734399, 10.79262260565454193959757688576
