| L(s) = 1 | + (−1.79 + 0.381i)2-s + (0.229 + 2.18i)3-s + (1.24 − 0.554i)4-s + (−0.425 + 0.472i)5-s + (−1.24 − 3.83i)6-s + (1.28 − 2.31i)7-s + (0.943 − 0.685i)8-s + (−1.79 + 0.381i)9-s + (0.582 − 1.00i)10-s + (1.5 + 2.59i)12-s + (−0.556 + 1.71i)13-s + (−1.42 + 4.63i)14-s + (−1.13 − 0.821i)15-s + (−3.25 + 3.61i)16-s + (2.77 + 0.589i)17-s + (3.07 − 1.36i)18-s + ⋯ |
| L(s) = 1 | + (−1.26 + 0.269i)2-s + (0.132 + 1.26i)3-s + (0.623 − 0.277i)4-s + (−0.190 + 0.211i)5-s + (−0.508 − 1.56i)6-s + (0.486 − 0.873i)7-s + (0.333 − 0.242i)8-s + (−0.598 + 0.127i)9-s + (0.184 − 0.319i)10-s + (0.433 + 0.750i)12-s + (−0.154 + 0.475i)13-s + (−0.381 + 1.23i)14-s + (−0.291 − 0.212i)15-s + (−0.814 + 0.904i)16-s + (0.672 + 0.142i)17-s + (0.724 − 0.322i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.212630 + 0.665116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.212630 + 0.665116i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.28 + 2.31i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (1.79 - 0.381i)T + (1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.229 - 2.18i)T + (-2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (0.425 - 0.472i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (0.556 - 1.71i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.77 - 0.589i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (5.08 + 2.26i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (1.08 + 1.87i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-8.43 - 6.13i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.30 - 4.77i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.634 - 6.03i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (6.09 - 4.42i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 + (2.58 + 1.15i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.99 - 5.55i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-10.7 + 4.80i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (2.89 - 3.21i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.801 + 1.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.32 - 4.08i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (14.6 - 6.50i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (4.65 - 0.990i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (2.85 + 8.77i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (0.182 + 0.315i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.802 - 2.46i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.15791824075280876276150064762, −9.962013267638764374473416735571, −8.609012792768719654368012313337, −8.509968965151833315633911132249, −7.20776225882423290137314043998, −6.65061336190685900122082556515, −4.88637147993941668675527333830, −4.34902537004730210659813830282, −3.24846367793746130617286157698, −1.34208098902817758246617985184,
0.57198372427450462986408896621, 1.82101246979081511950623441077, 2.57307794461316510161264592499, 4.47244149507561208662444005969, 5.70380853430555997903430138524, 6.67370944247182064623221099378, 7.73730775180634098691514510424, 8.243339919242699680298028157009, 8.645911027785631312963378419840, 9.882915902271683968333253841406
