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
−10.63630387911912181773988258494, −9.751851925078572162806105002815, −9.230373172929313786351555498960, −8.310454250236823503039985637626, −7.29705622287925794335586195555, −6.14826697690942505754480098899, −5.38248650145042078927120393561, −4.20613497349120786563186883738, −3.23459431344796938251495829538, −2.03776689283180431170308530933,
1.53161774258381377001692278083, 2.74931101228654801156853980126, 3.47593807575157898253752504444, 4.87503885947787488106672347469, 6.21047793930950795955488777366, 6.82923334567092311421209762694, 7.73189626091782085775257237377, 9.201243009268572241479334045747, 9.760478628799739763113485759723, 10.29115833436281668220018437514