| L(s) = 1 | + (0.0771 − 1.41i)2-s + (−0.342 + 0.939i)3-s + (−1.98 − 0.217i)4-s + (1.02 − 0.181i)5-s + (1.30 + 0.555i)6-s + (1.05 − 1.83i)7-s + (−0.460 + 2.79i)8-s + (−0.766 − 0.642i)9-s + (−0.176 − 1.46i)10-s + (−0.636 + 0.367i)11-s + (0.884 − 1.79i)12-s + (−1.39 − 3.83i)13-s + (−2.50 − 1.63i)14-s + (−0.181 + 1.02i)15-s + (3.90 + 0.866i)16-s + (5.40 − 4.53i)17-s + ⋯ |
| L(s) = 1 | + (0.0545 − 0.998i)2-s + (−0.197 + 0.542i)3-s + (−0.994 − 0.108i)4-s + (0.459 − 0.0809i)5-s + (0.530 + 0.226i)6-s + (0.399 − 0.692i)7-s + (−0.162 + 0.986i)8-s + (−0.255 − 0.214i)9-s + (−0.0558 − 0.463i)10-s + (−0.191 + 0.110i)11-s + (0.255 − 0.517i)12-s + (−0.386 − 1.06i)13-s + (−0.669 − 0.436i)14-s + (−0.0467 + 0.265i)15-s + (0.976 + 0.216i)16-s + (1.31 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.801029 - 1.00148i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.801029 - 1.00148i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0771 + 1.41i)T \) |
| 3 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (-4.25 - 0.939i)T \) |
| good | 5 | \( 1 + (-1.02 + 0.181i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.05 + 1.83i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.636 - 0.367i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.39 + 3.83i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-5.40 + 4.53i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.20 + 6.85i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.88 + 4.63i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.59 - 7.96i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.26iT - 37T^{2} \) |
| 41 | \( 1 + (9.52 + 3.46i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (5.13 - 0.905i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.41 - 3.70i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-5.68 - 1.00i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.657 + 0.783i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-14.7 - 2.59i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.24 - 8.63i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.60 - 9.10i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.11 - 0.406i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.46 - 0.533i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.62 - 3.82i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.20 + 3.35i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (9.62 - 8.07i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.51769454205174046635293826108, −10.22407277487288613602525353576, −9.408955919743258954436721682290, −8.290753357400743600614127326163, −7.30438352393860775485408008910, −5.51337695858530553964208787582, −5.04804437316730602861616989167, −3.76514752482073160464623202185, −2.68598102389976064722137175530, −0.894700537024343943796154673964,
1.68016869747385166032555888397, 3.51713103056005234954425703254, 5.11466197331167465787999322808, 5.68452957380040970803460822078, 6.66704105085654834384021052573, 7.61490718446848019861390722252, 8.379331706369259077921876100476, 9.399863159054804794629673135167, 10.12778993094934857762860317408, 11.66751598001411385542640900457
