L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−1.04 − 1.43i)5-s + (−0.951 + 0.309i)7-s + (0.309 − 0.951i)8-s + 1.77i·10-s + (−0.808 + 3.21i)11-s + (−2.87 + 3.95i)13-s + (0.951 + 0.309i)14-s + (−0.809 + 0.587i)16-s + (3.29 − 2.39i)17-s + (6.36 + 2.06i)19-s + (1.04 − 1.43i)20-s + (2.54 − 2.12i)22-s − 8.52i·23-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.466 − 0.641i)5-s + (−0.359 + 0.116i)7-s + (0.109 − 0.336i)8-s + 0.561i·10-s + (−0.243 + 0.969i)11-s + (−0.796 + 1.09i)13-s + (0.254 + 0.0825i)14-s + (−0.202 + 0.146i)16-s + (0.798 − 0.580i)17-s + (1.46 + 0.474i)19-s + (0.233 − 0.320i)20-s + (0.542 − 0.453i)22-s − 1.77i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6675382484\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6675382484\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.951 - 0.309i)T \) |
| 11 | \( 1 + (0.808 - 3.21i)T \) |
good | 5 | \( 1 + (1.04 + 1.43i)T + (-1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (2.87 - 3.95i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.29 + 2.39i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-6.36 - 2.06i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 8.52iT - 23T^{2} \) |
| 29 | \( 1 + (1.42 + 4.38i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (5.66 + 4.11i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.200 - 0.617i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.28 + 10.1i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.80iT - 43T^{2} \) |
| 47 | \( 1 + (1.05 + 0.344i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.49 - 7.55i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.65 - 1.18i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (5.09 + 7.01i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 6.30T + 67T^{2} \) |
| 71 | \( 1 + (0.602 + 0.829i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.69 - 1.20i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.18 + 12.6i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (2.47 - 1.79i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 10.7iT - 89T^{2} \) |
| 97 | \( 1 + (-0.363 - 0.264i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.511277570783984860998484129255, −8.635033888483751025702468003080, −7.58208112348201741729369106922, −7.28050308521556917048912893136, −6.03966042111624470045903060012, −4.83400806952952141475307468660, −4.20062404528835276196275446904, −2.92987270478042728378885495296, −1.87969934579015172149908178614, −0.36675621717364068283179069244,
1.19808934735798465506426036866, 3.08843199610959076349667090523, 3.41645726614421900784488921845, 5.25015403683486460287145650530, 5.64398148565916924025873233547, 6.84104361559131662305207348348, 7.60458926766387045342894281665, 7.923638074897327150427446445773, 9.148083778291433717576021671769, 9.741660472263690997222139432724