L(s) = 1 | + (−0.866 + 1.5i)2-s + (−0.5 − 0.866i)4-s + (0.866 + 1.5i)5-s + (−1 + 1.73i)7-s − 1.73·8-s − 3·10-s + (1.73 − 3i)11-s + (0.5 + 0.866i)13-s + (−1.73 − 3i)14-s + (2.49 − 4.33i)16-s + 5.19·17-s + 2·19-s + (0.866 − 1.5i)20-s + (3 + 5.19i)22-s + (−1.73 − 3i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 1.06i)2-s + (−0.250 − 0.433i)4-s + (0.387 + 0.670i)5-s + (−0.377 + 0.654i)7-s − 0.612·8-s − 0.948·10-s + (0.522 − 0.904i)11-s + (0.138 + 0.240i)13-s + (−0.462 − 0.801i)14-s + (0.624 − 1.08i)16-s + 1.26·17-s + 0.458·19-s + (0.193 − 0.335i)20-s + (0.639 + 1.10i)22-s + (−0.361 − 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.426089 + 0.608519i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.426089 + 0.608519i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.866 - 1.5i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.73 + 3i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.866 - 1.5i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (-3.46 - 6i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.46 - 6i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (6.92 + 12i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.92 + 12i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.72694841325430609119950930082, −14.13475353177627978308075222494, −12.53381154811084895897037443824, −11.40095851711680885976908215935, −9.905815052488909485105400398566, −8.946766933064434179955492562636, −7.80685317461636902016303167721, −6.49233329593592705810758932099, −5.76005416609235568050894748916, −3.13565787939421405824023771537,
1.45663024891825141509450155705, 3.55736496586257136563323663316, 5.49826976682736586324062303365, 7.22811549961406415405763942158, 8.861804788918030230872677075765, 9.778329325714545354326930716590, 10.49906119444550916472184614542, 11.91364783632519532226551784387, 12.59757293128298309765001418770, 13.79870726771932030864587185086