L(s) = 1 | + (−0.334 − 0.192i)2-s + (0.139 + 1.72i)3-s + (−0.925 − 1.60i)4-s + (0.286 − 0.603i)6-s + (−2.36 + 1.17i)7-s + 1.48i·8-s + (−2.96 + 0.480i)9-s + (2.20 − 1.27i)11-s + (2.63 − 1.82i)12-s − 3.06i·13-s + (1.01 + 0.0640i)14-s + (−1.56 + 2.71i)16-s + (−3.23 − 5.59i)17-s + (1.08 + 0.410i)18-s + (−1.03 − 0.597i)19-s + ⋯ |
L(s) = 1 | + (−0.236 − 0.136i)2-s + (0.0803 + 0.996i)3-s + (−0.462 − 0.801i)4-s + (0.116 − 0.246i)6-s + (−0.895 + 0.444i)7-s + 0.525i·8-s + (−0.987 + 0.160i)9-s + (0.663 − 0.383i)11-s + (0.761 − 0.525i)12-s − 0.850i·13-s + (0.272 + 0.0171i)14-s + (−0.391 + 0.677i)16-s + (−0.783 − 1.35i)17-s + (0.254 + 0.0968i)18-s + (−0.237 − 0.137i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.166831 - 0.329212i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.166831 - 0.329212i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.139 - 1.72i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.36 - 1.17i)T \) |
good | 2 | \( 1 + (0.334 + 0.192i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.20 + 1.27i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.06iT - 13T^{2} \) |
| 17 | \( 1 + (3.23 + 5.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.03 + 0.597i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.64 + 1.52i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.77iT - 29T^{2} \) |
| 31 | \( 1 + (5.95 - 3.43i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.77 - 3.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 + (1.61 - 2.78i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.4 + 6.62i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.98 - 3.43i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.08 + 4.67i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.75 + 3.04i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 0.921iT - 71T^{2} \) |
| 73 | \( 1 + (0.256 - 0.148i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.14 - 7.18i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.11T + 83T^{2} \) |
| 89 | \( 1 + (9.41 - 16.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.3iT - 97T^{2} \) |
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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
−10.32688898181906016459793305635, −9.659088836040314992484239336019, −9.061434189762599569556093359132, −8.313945439500158277449099453344, −6.64000993144877388479602248208, −5.74308687827467107866988524305, −4.91435567590478837886438148735, −3.75590373970217208145125122854, −2.53393795983369730489632662016, −0.22480929405697082734978097949,
1.81293723027543915761514149399, 3.41783692477824567712954318074, 4.23308918849880942085159989703, 5.98949096648217939574206626865, 6.89972936635025021202644457828, 7.37145349934355974883807226632, 8.637579825750240660450550121193, 9.027463713057707018894130578254, 10.15789726759036361307970259532, 11.35201293488656747838047373154