| L(s) = 1 | + (−0.847 + 1.13i)2-s + (−0.462 + 1.66i)3-s + (−0.562 − 1.91i)4-s + (0.539 + 0.354i)5-s + (−1.49 − 1.93i)6-s + (0.568 − 0.536i)7-s + (2.64 + 0.989i)8-s + (−2.57 − 1.54i)9-s + (−0.858 + 0.309i)10-s + (2.27 − 4.53i)11-s + (3.46 − 0.0527i)12-s + (2.07 + 0.893i)13-s + (0.125 + 1.09i)14-s + (−0.841 + 0.736i)15-s + (−3.36 + 2.16i)16-s + (3.54 − 4.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.599 + 0.800i)2-s + (−0.266 + 0.963i)3-s + (−0.281 − 0.959i)4-s + (0.241 + 0.158i)5-s + (−0.611 − 0.791i)6-s + (0.214 − 0.202i)7-s + (0.936 + 0.349i)8-s + (−0.857 − 0.514i)9-s + (−0.271 + 0.0979i)10-s + (0.686 − 1.36i)11-s + (0.999 − 0.0152i)12-s + (0.574 + 0.247i)13-s + (0.0334 + 0.293i)14-s + (−0.217 + 0.190i)15-s + (−0.841 + 0.540i)16-s + (0.860 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.966048 + 0.471968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.966048 + 0.471968i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.847 - 1.13i)T \) |
| 3 | \( 1 + (0.462 - 1.66i)T \) |
| good | 5 | \( 1 + (-0.539 - 0.354i)T + (1.98 + 4.59i)T^{2} \) |
| 7 | \( 1 + (-0.568 + 0.536i)T + (0.407 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-2.27 + 4.53i)T + (-6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (-2.07 - 0.893i)T + (8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (-3.54 + 4.22i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (0.124 - 0.104i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-1.52 + 1.61i)T + (-1.33 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-8.67 - 1.01i)T + (28.2 + 6.68i)T^{2} \) |
| 31 | \( 1 + (1.62 - 6.87i)T + (-27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (3.35 + 0.592i)T + (34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (-2.81 + 2.09i)T + (11.7 - 39.2i)T^{2} \) |
| 43 | \( 1 + (0.158 + 2.72i)T + (-42.7 + 4.99i)T^{2} \) |
| 47 | \( 1 + (10.9 - 2.59i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (-4.44 + 7.69i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.04 - 8.06i)T + (-35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-14.6 - 4.39i)T + (50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (9.65 - 1.12i)T + (65.1 - 15.4i)T^{2} \) |
| 71 | \( 1 + (10.2 - 3.73i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (1.89 + 0.688i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (0.407 + 0.303i)T + (22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (-0.385 - 0.287i)T + (23.8 + 79.5i)T^{2} \) |
| 89 | \( 1 + (3.27 - 9.01i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-8.36 + 5.50i)T + (38.4 - 89.0i)T^{2} \) |
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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
−10.44336518503372293887712464516, −9.833408144746463851369949898985, −8.734472447447499365483759839782, −8.480982072151956675775244663347, −7.01037945393460823833391890852, −6.17258363135541423645382634109, −5.40589133004793694046345240936, −4.39619256028789431196521170583, −3.14536876402565167247826453423, −0.914430389794917158655228585427,
1.25820250116494721680675383358, 2.08483696190375434441675126190, 3.51953299116209813236499216980, 4.83339612150074546718166127902, 6.07623716314259298248657713231, 7.09907822395471880626387172131, 7.929064323178586088923680043150, 8.660457934068759415544794052275, 9.671954600939725208491639358551, 10.39968203669553392111397772365
