L(s) = 1 | + (0.5 − 0.866i)2-s + (0.320 − 1.70i)3-s + (−0.499 − 0.866i)4-s + (1.62 − 0.936i)5-s + (−1.31 − 1.12i)6-s + (2.47 + 0.928i)7-s − 0.999·8-s + (−2.79 − 1.08i)9-s − 1.87i·10-s + 5.09·11-s + (−1.63 + 0.573i)12-s + (3.20 + 1.64i)13-s + (2.04 − 1.68i)14-s + (−1.07 − 3.06i)15-s + (−0.5 + 0.866i)16-s + (−1.48 − 2.56i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.184 − 0.982i)3-s + (−0.249 − 0.433i)4-s + (0.725 − 0.418i)5-s + (−0.536 − 0.460i)6-s + (0.936 + 0.350i)7-s − 0.353·8-s + (−0.931 − 0.363i)9-s − 0.592i·10-s + 1.53·11-s + (−0.471 + 0.165i)12-s + (0.889 + 0.456i)13-s + (0.545 − 0.449i)14-s + (−0.277 − 0.790i)15-s + (−0.125 + 0.216i)16-s + (−0.359 − 0.622i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24659 - 1.84298i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24659 - 1.84298i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.320 + 1.70i)T \) |
| 7 | \( 1 + (-2.47 - 0.928i)T \) |
| 13 | \( 1 + (-3.20 - 1.64i)T \) |
good | 5 | \( 1 + (-1.62 + 0.936i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 5.09T + 11T^{2} \) |
| 17 | \( 1 + (1.48 + 2.56i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 + (5.25 + 3.03i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (7.30 - 4.21i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.09 - 7.09i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.52 - 3.18i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.85 + 2.80i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.87 - 8.44i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.70 + 1.56i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.21 + 4.74i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.90 - 1.10i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 5.90iT - 61T^{2} \) |
| 67 | \( 1 + 11.0iT - 67T^{2} \) |
| 71 | \( 1 + (-7.27 + 12.6i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.99 - 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.14 + 8.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.63iT - 83T^{2} \) |
| 89 | \( 1 + (-12.7 - 7.38i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.83 - 10.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.93719594885047786622211472378, −9.286224423303540816216501750979, −9.027049439060753514751239916849, −7.992334502307415407905970321571, −6.63191803636985034299935971153, −5.99043828218718572054181738910, −4.86044382038371304567088817348, −3.62479514201048453371225026725, −1.99224025325589376840773272718, −1.42499978070207048756018914816,
2.04209459794585230446882930578, 3.87593645358775296255209857149, 4.18501580103901002075761536795, 5.78284348085469990989442840311, 6.09762805870445736753488775793, 7.55149041478972032147852394933, 8.459203463539079333766316313765, 9.270303593875117938749454933342, 10.11124567508984114474612479684, 11.11877387898562288205664550617