| L(s) = 1 | + (0.830 − 2.40i)2-s + (0.458 + 0.888i)3-s + (−3.49 − 2.75i)4-s + (3.07 − 0.293i)5-s + (2.51 − 0.361i)6-s + (−2.46 − 0.973i)7-s + (−5.23 + 3.36i)8-s + (−0.580 + 0.814i)9-s + (1.84 − 7.62i)10-s + (4.67 − 1.61i)11-s + (0.842 − 4.37i)12-s + (−1.23 − 4.20i)13-s + (−4.38 + 5.09i)14-s + (1.67 + 2.59i)15-s + (1.62 + 6.71i)16-s + (0.915 + 0.366i)17-s + ⋯ |
| L(s) = 1 | + (0.587 − 1.69i)2-s + (0.264 + 0.513i)3-s + (−1.74 − 1.37i)4-s + (1.37 − 0.131i)5-s + (1.02 − 0.147i)6-s + (−0.929 − 0.367i)7-s + (−1.85 + 1.19i)8-s + (−0.193 + 0.271i)9-s + (0.584 − 2.41i)10-s + (1.40 − 0.487i)11-s + (0.243 − 1.26i)12-s + (−0.342 − 1.16i)13-s + (−1.17 + 1.36i)14-s + (0.431 + 0.671i)15-s + (0.407 + 1.67i)16-s + (0.222 + 0.0889i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.702435 - 2.02546i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.702435 - 2.02546i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.458 - 0.888i)T \) |
| 7 | \( 1 + (2.46 + 0.973i)T \) |
| 23 | \( 1 + (-4.78 - 0.284i)T \) |
| good | 2 | \( 1 + (-0.830 + 2.40i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (-3.07 + 0.293i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-4.67 + 1.61i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (1.23 + 4.20i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.915 - 0.366i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (5.65 - 2.26i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (0.209 + 1.45i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-1.21 + 0.0578i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-8.71 - 6.20i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (3.66 + 1.67i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (4.22 - 6.56i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (6.71 - 3.87i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.13 - 6.43i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-5.20 - 1.26i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-3.00 - 1.55i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (1.52 + 7.92i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (-6.02 - 6.94i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.22 + 1.56i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (6.34 - 6.65i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-4.44 - 9.74i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.646 + 13.5i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (3.90 - 8.55i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.53088167199915268909931381884, −9.857940267511497577040273704686, −9.499771535105981768159972691862, −8.512321080983267699626030067625, −6.45246124331233907238687399235, −5.66210116510601462741067661255, −4.49009777935108962749663313817, −3.45977302604571631541268641468, −2.61321567119505405806819257005, −1.19407489736053959337390252123,
2.18080985279111587086361036266, 3.82093947226421980824834317992, 5.02597838678209436146018389063, 6.17852565641139850472081285219, 6.61800498894132628084324085270, 7.10760825457630522863353337243, 8.730425551555742077550287408185, 9.135159615551762151187517854954, 9.915657293090291192801925090520, 11.69302861438591815945565635845
