| L(s) = 1 | + (−0.430 + 1.34i)2-s + (2.54 + 0.250i)3-s + (−1.62 − 1.16i)4-s + (0.471 + 0.881i)5-s + (−1.43 + 3.32i)6-s + (0.101 + 0.511i)7-s + (2.26 − 1.69i)8-s + (3.47 + 0.691i)9-s + (−1.39 + 0.254i)10-s + (1.61 + 1.32i)11-s + (−3.85 − 3.36i)12-s + (1.25 + 0.668i)13-s + (−0.733 − 0.0834i)14-s + (0.979 + 2.36i)15-s + (1.30 + 3.78i)16-s + (0.336 − 0.813i)17-s + ⋯ |
| L(s) = 1 | + (−0.304 + 0.952i)2-s + (1.47 + 0.144i)3-s + (−0.814 − 0.580i)4-s + (0.210 + 0.394i)5-s + (−0.585 + 1.35i)6-s + (0.0384 + 0.193i)7-s + (0.801 − 0.598i)8-s + (1.15 + 0.230i)9-s + (−0.439 + 0.0806i)10-s + (0.486 + 0.398i)11-s + (−1.11 − 0.971i)12-s + (0.346 + 0.185i)13-s + (−0.196 − 0.0223i)14-s + (0.252 + 0.610i)15-s + (0.326 + 0.945i)16-s + (0.0817 − 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0413 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0413 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42157 + 1.48158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42157 + 1.48158i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.430 - 1.34i)T \) |
| 5 | \( 1 + (-0.471 - 0.881i)T \) |
| good | 3 | \( 1 + (-2.54 - 0.250i)T + (2.94 + 0.585i)T^{2} \) |
| 7 | \( 1 + (-0.101 - 0.511i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.61 - 1.32i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 0.668i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.336 + 0.813i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.89 + 0.575i)T + (15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (0.408 - 0.272i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.0332 - 0.0405i)T + (-5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (2.05 - 2.05i)T - 31iT^{2} \) |
| 37 | \( 1 + (2.50 - 8.24i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (2.42 + 3.63i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-4.69 + 0.462i)T + (42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (5.31 + 2.20i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (1.47 - 1.79i)T + (-10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-8.74 + 4.67i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-1.06 + 10.7i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (0.232 - 2.35i)T + (-65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (5.15 - 1.02i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (0.318 - 1.59i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (8.74 - 3.62i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (3.04 + 10.0i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (4.60 + 3.07i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-4.36 + 4.36i)T - 97iT^{2} \) |
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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
−10.27457531120385748183964265201, −9.636363752411702035251593756246, −8.903884306117211589293594090873, −8.287426050354932843554722481132, −7.34597021297849695068464269776, −6.64837481815486719576198950141, −5.43100908056766316094277729003, −4.21496599629199502561887345522, −3.19990659691631957900047800963, −1.75700894732252806531789745387,
1.25981177212885060755038172584, 2.42014453489185420158943865717, 3.47072338654195667357634583029, 4.21375709655545429339887965813, 5.65450623518057055463639844928, 7.26711909142552409252610965846, 8.087388409928713530699745313189, 8.793877947873469142977567520718, 9.336738046229279761609180320604, 10.16969861927396533956113418267
