| L(s) = 1 | + 2-s − 3·4-s + 1.73i·5-s + (−3.5 + 6.06i)7-s − 7·8-s + 1.73i·10-s − 17·11-s − 17.3i·13-s + (−3.5 + 6.06i)14-s + 5·16-s + 31.1i·17-s + 20.7i·19-s − 5.19i·20-s − 17·22-s − 32·23-s + ⋯ |
| L(s) = 1 | + 0.5·2-s − 0.750·4-s + 0.346i·5-s + (−0.5 + 0.866i)7-s − 0.875·8-s + 0.173i·10-s − 1.54·11-s − 1.33i·13-s + (−0.250 + 0.433i)14-s + 0.312·16-s + 1.83i·17-s + 1.09i·19-s − 0.259i·20-s − 0.772·22-s − 1.39·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.149007 + 0.556104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149007 + 0.556104i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (3.5 - 6.06i)T \) |
| good | 2 | \( 1 - T + 4T^{2} \) |
| 5 | \( 1 - 1.73iT - 25T^{2} \) |
| 11 | \( 1 + 17T + 121T^{2} \) |
| 13 | \( 1 + 17.3iT - 169T^{2} \) |
| 17 | \( 1 - 31.1iT - 289T^{2} \) |
| 19 | \( 1 - 20.7iT - 361T^{2} \) |
| 23 | \( 1 + 32T + 529T^{2} \) |
| 29 | \( 1 + 2T + 841T^{2} \) |
| 31 | \( 1 + 22.5iT - 961T^{2} \) |
| 37 | \( 1 - 8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 26T + 1.84e3T^{2} \) |
| 47 | \( 1 + 38.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - T + 2.80e3T^{2} \) |
| 59 | \( 1 - 90.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 69.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 34T + 4.48e3T^{2} \) |
| 71 | \( 1 - 10T + 5.04e3T^{2} \) |
| 73 | \( 1 - 57.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 46T + 6.24e3T^{2} \) |
| 83 | \( 1 - 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 51.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 91.7iT - 9.40e3T^{2} \) |
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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
−12.77116681215677246141968053360, −12.23117804345371914011053653621, −10.51929988321912560901238440231, −10.02381456096995340894473142818, −8.541791010621970061454673428167, −7.914037290470480177414403438918, −5.93871724284969019484933223568, −5.54032767006235777613325884001, −3.88813747252085528635158596070, −2.68260966030812531205220731248,
0.27855393976472914367043885314, 2.91287068670085263631339624691, 4.42068306404786360271414731977, 5.11864176104495110377628319430, 6.64970874101008954596164589703, 7.78231874251645278760711753648, 9.095117619245174764350320389155, 9.785918865459859663624757060411, 10.99806804712751197822960994793, 12.17280929211006599713793801629
