| L(s) = 1 | + (0.819 − 0.573i)2-s + (1.50 − 0.855i)3-s + (0.342 − 0.939i)4-s + (1.97 + 1.04i)5-s + (0.743 − 1.56i)6-s + (−2.09 − 0.562i)7-s + (−0.258 − 0.965i)8-s + (1.53 − 2.57i)9-s + (2.21 − 0.282i)10-s + (2.25 + 1.30i)11-s + (−0.288 − 1.70i)12-s + (−2.88 − 0.252i)13-s + (−2.04 + 0.742i)14-s + (3.87 − 0.123i)15-s + (−0.766 − 0.642i)16-s + (1.92 − 1.34i)17-s + ⋯ |
| L(s) = 1 | + (0.579 − 0.405i)2-s + (0.869 − 0.493i)3-s + (0.171 − 0.469i)4-s + (0.884 + 0.465i)5-s + (0.303 − 0.638i)6-s + (−0.793 − 0.212i)7-s + (−0.0915 − 0.341i)8-s + (0.512 − 0.858i)9-s + (0.701 − 0.0892i)10-s + (0.679 + 0.392i)11-s + (−0.0832 − 0.493i)12-s + (−0.801 − 0.0700i)13-s + (−0.545 + 0.198i)14-s + (0.999 − 0.0320i)15-s + (−0.191 − 0.160i)16-s + (0.465 − 0.326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.43577 - 1.44358i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43577 - 1.44358i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.819 + 0.573i)T \) |
| 3 | \( 1 + (-1.50 + 0.855i)T \) |
| 5 | \( 1 + (-1.97 - 1.04i)T \) |
| 19 | \( 1 + (-3.59 + 2.46i)T \) |
| good | 7 | \( 1 + (2.09 + 0.562i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.25 - 1.30i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.88 + 0.252i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.92 + 1.34i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (5.73 - 2.67i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (0.986 + 5.59i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.45 - 7.71i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.53 - 8.53i)T - 37iT^{2} \) |
| 41 | \( 1 + (3.23 - 3.86i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-7.06 - 3.29i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-1.86 + 2.65i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (7.30 - 3.40i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (0.302 - 1.71i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (6.68 + 2.43i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.51 + 1.05i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (4.61 + 12.6i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.25 - 14.3i)T + (-71.8 + 12.6i)T^{2} \) |
| 79 | \( 1 + (-7.19 + 8.57i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.93 - 1.85i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.349 + 0.293i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.74 + 2.48i)T + (-33.1 + 91.1i)T^{2} \) |
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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
−10.29115833436281668220018437514, −9.760478628799739763113485759723, −9.201243009268572241479334045747, −7.73189626091782085775257237377, −6.82923334567092311421209762694, −6.21047793930950795955488777366, −4.87503885947787488106672347469, −3.47593807575157898253752504444, −2.74931101228654801156853980126, −1.53161774258381377001692278083,
2.03776689283180431170308530933, 3.23459431344796938251495829538, 4.20613497349120786563186883738, 5.38248650145042078927120393561, 6.14826697690942505754480098899, 7.29705622287925794335586195555, 8.310454250236823503039985637626, 9.230373172929313786351555498960, 9.751851925078572162806105002815, 10.63630387911912181773988258494
