L(s) = 1 | − 2-s + 1.73i·3-s + 4-s + (1.5 + 2.59i)5-s − 1.73i·6-s − 8-s − 2.99·9-s + (−1.5 − 2.59i)10-s + (1.5 − 2.59i)11-s + 1.73i·12-s + (−0.5 + 0.866i)13-s + (−4.5 + 2.59i)15-s + 16-s + (1.5 + 2.59i)17-s + 2.99·18-s + (−3.5 + 6.06i)19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.999i·3-s + 0.5·4-s + (0.670 + 1.16i)5-s − 0.707i·6-s − 0.353·8-s − 0.999·9-s + (−0.474 − 0.821i)10-s + (0.452 − 0.783i)11-s + 0.499i·12-s + (−0.138 + 0.240i)13-s + (−1.16 + 0.670i)15-s + 0.250·16-s + (0.363 + 0.630i)17-s + 0.707·18-s + (−0.802 + 1.39i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.245382 + 1.01188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.245382 + 1.01188i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.39562127406394720345447818060, −9.729209829164765977743811430845, −9.057205605416533279803614600937, −8.137300721426646937940238490548, −7.11108174311981492139832303749, −6.04688242461541621953646227894, −5.60617276254181397099908486755, −3.87426043143048719017810797897, −3.17609009421795780705131170886, −1.86662004822149684102865916908,
0.62175685291960500621511822822, 1.74079645369750714462172359529, 2.74438599493495838160807960792, 4.65000065925864479939546635242, 5.48515262991420957884108594787, 6.62053404230005211540171888006, 7.17869515237098545801502589486, 8.230039808944002442357392488973, 9.063372790816131529363498390117, 9.330575465001104408978983883949