L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.485 − 2.75i)3-s + (0.766 − 0.642i)4-s + (−1.79 − 1.50i)5-s + (0.485 + 2.75i)6-s + (0.642 + 1.11i)7-s + (−0.500 + 0.866i)8-s + (−4.51 − 1.64i)9-s + (2.20 + 0.801i)10-s + (−2.87 + 4.98i)11-s + (−1.39 − 2.41i)12-s + (0.0528 + 0.299i)13-s + (−0.983 − 0.825i)14-s + (−5.01 + 4.21i)15-s + (0.173 − 0.984i)16-s + (−3.93 + 1.43i)17-s + ⋯ |
L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.280 − 1.58i)3-s + (0.383 − 0.321i)4-s + (−0.803 − 0.673i)5-s + (0.198 + 1.12i)6-s + (0.242 + 0.420i)7-s + (−0.176 + 0.306i)8-s + (−1.50 − 0.547i)9-s + (0.696 + 0.253i)10-s + (−0.867 + 1.50i)11-s + (−0.403 − 0.698i)12-s + (0.0146 + 0.0831i)13-s + (−0.262 − 0.220i)14-s + (−1.29 + 1.08i)15-s + (0.0434 − 0.246i)16-s + (−0.954 + 0.347i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0356241 + 0.0475842i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0356241 + 0.0475842i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.485 + 2.75i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (1.79 + 1.50i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.642 - 1.11i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.87 - 4.98i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0528 - 0.299i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.93 - 1.43i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (4.96 - 4.16i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.93 - 1.06i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.22 + 5.57i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.97T + 37T^{2} \) |
| 41 | \( 1 + (0.871 - 4.94i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.757 + 0.635i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.12 - 1.50i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.52 + 2.11i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (3.11 - 1.13i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.39 - 7.04i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.11 - 1.49i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (3.38 + 2.83i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.393 - 2.23i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.48 + 8.44i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.88 - 8.45i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.61 + 14.8i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (16.3 - 5.93i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.66016368960707772577809127211, −9.533366707969648531177267764212, −8.588450415075703775732541807072, −7.961780941937626088078750182219, −7.44994833172846980233769873856, −6.63685580232745195144864647947, −5.51733784212653874112178944836, −4.31371537067001861890836803651, −2.43517192385054151910659008974, −1.63880664823946982169124090879,
0.03518258495033353923458803141, 2.71306603554090351487367363830, 3.50615254352846147202254980017, 4.33135785945573145672596398487, 5.50078365762029883313384732712, 6.80672409886032364096099762473, 7.946001853763672742888052742165, 8.533795679248643692830884897865, 9.314415518808728531562983716005, 10.54832128200615291331725708627