L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.880 + 1.52i)5-s + (2.56 − 0.658i)7-s − 0.999·8-s + (−0.880 + 1.52i)10-s + (3.06 − 5.30i)11-s + 0.760·13-s + (1.85 + 1.88i)14-s + (−0.5 − 0.866i)16-s + (3.42 − 5.92i)17-s + (0.971 + 1.68i)19-s − 1.76·20-s + 6.12·22-s + (−0.210 − 0.364i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.393 + 0.681i)5-s + (0.968 − 0.249i)7-s − 0.353·8-s + (−0.278 + 0.482i)10-s + (0.923 − 1.59i)11-s + 0.211·13-s + (0.494 + 0.505i)14-s + (−0.125 − 0.216i)16-s + (0.829 − 1.43i)17-s + (0.222 + 0.385i)19-s − 0.393·20-s + 1.30·22-s + (−0.0438 − 0.0760i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.436350050\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.436350050\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.56 + 0.658i)T \) |
good | 5 | \( 1 + (-0.880 - 1.52i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.06 + 5.30i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.760T + 13T^{2} \) |
| 17 | \( 1 + (-3.42 + 5.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.971 - 1.68i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.210 + 0.364i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 + (3.85 - 6.67i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 + (-0.830 - 1.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.112 + 0.195i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.993 + 1.72i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.17 - 8.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.39 - 5.87i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + (-0.153 + 0.265i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.72 - 11.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.63T + 97T^{2} \) |
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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
−9.921497283400957197132213759035, −8.814914029912856319780932789102, −8.283904402364239237912381906309, −7.26151252620628026419661055910, −6.58400647022600455545709098858, −5.68458322720539355061254974141, −4.95042465566032147423620772093, −3.70181631041236373328971388603, −2.91067094641259578958686924251, −1.19271664493966280017416656540,
1.44979093194400235010572495937, 1.96404690994354780882859979109, 3.63938356121400714879875100340, 4.53252501907953445090545105112, 5.21689185501827017839741352555, 6.13043938737937165302244007878, 7.27050268492457246879618855024, 8.248042249337915885439188774576, 9.069131766374499481897233377279, 9.752024688323455469160190880670