L(s) = 1 | + 2-s + (−0.866 − 0.5i)3-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + (−0.5 − 0.866i)7-s − 8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)10-s − 11-s + (−0.866 − 0.5i)13-s + (−0.5 − 0.866i)14-s + 0.999·15-s − 16-s + (−0.866 − 0.5i)17-s + (0.499 + 0.866i)18-s + (0.866 + 0.5i)19-s + ⋯ |
L(s) = 1 | + 2-s + (−0.866 − 0.5i)3-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + (−0.5 − 0.866i)7-s − 8-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)10-s − 11-s + (−0.866 − 0.5i)13-s + (−0.5 − 0.866i)14-s + 0.999·15-s − 16-s + (−0.866 − 0.5i)17-s + (0.499 + 0.866i)18-s + (0.866 + 0.5i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1931764643\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1931764643\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
good | 2 | \( 1 - T + T^{2} \) |
| 5 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + iT - T^{2} \) |
| 61 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−10.33686879352744536556787105350, −9.371038350033295547952915098548, −7.73394762308907324783502833770, −7.44910716223101986076683756530, −6.43491238309343556441350916157, −5.46386461491144483489677585181, −4.69201411489182634797521909661, −3.73905880392271394081806112982, −2.67815917289982455511262941580, −0.15027073046117490147359709343,
2.72724094164769865293313061415, 3.86344837400627276332761803154, 4.75178576327852721300233440380, 5.25353406746247396608602035882, 6.16471034727461438207079867641, 7.15413363094971183052785154253, 8.454636224742442380802992411082, 9.227683338066229580044824282823, 10.01211022351571486964089994832, 11.16046666644785936458245698168