| L(s) = 1 | + (−1.99 + 0.180i)2-s + (1.05 + 0.862i)3-s + (3.93 − 0.718i)4-s + (−7.90 + 2.39i)5-s + (−2.24 − 1.52i)6-s + (1.29 − 6.52i)7-s + (−7.70 + 2.14i)8-s + (−1.39 − 7.01i)9-s + (15.3 − 6.19i)10-s + (0.0388 − 0.394i)11-s + (4.75 + 2.63i)12-s + (−1.94 + 6.42i)13-s + (−1.40 + 13.2i)14-s + (−10.3 − 4.29i)15-s + (14.9 − 5.65i)16-s + (−11.7 − 28.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.995 + 0.0901i)2-s + (0.350 + 0.287i)3-s + (0.983 − 0.179i)4-s + (−1.58 + 0.479i)5-s + (−0.374 − 0.254i)6-s + (0.185 − 0.932i)7-s + (−0.963 + 0.267i)8-s + (−0.154 − 0.779i)9-s + (1.53 − 0.619i)10-s + (0.00353 − 0.0358i)11-s + (0.396 + 0.219i)12-s + (−0.149 + 0.493i)13-s + (−0.100 + 0.945i)14-s + (−0.691 − 0.286i)15-s + (0.935 − 0.353i)16-s + (−0.691 − 1.67i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.183537 - 0.308301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.183537 - 0.308301i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 - 0.180i)T \) |
| good | 3 | \( 1 + (-1.05 - 0.862i)T + (1.75 + 8.82i)T^{2} \) |
| 5 | \( 1 + (7.90 - 2.39i)T + (20.7 - 13.8i)T^{2} \) |
| 7 | \( 1 + (-1.29 + 6.52i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-0.0388 + 0.394i)T + (-118. - 23.6i)T^{2} \) |
| 13 | \( 1 + (1.94 - 6.42i)T + (-140. - 93.8i)T^{2} \) |
| 17 | \( 1 + (11.7 + 28.3i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (7.55 + 14.1i)T + (-200. + 300. i)T^{2} \) |
| 23 | \( 1 + (19.3 + 12.8i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-2.58 - 26.2i)T + (-824. + 164. i)T^{2} \) |
| 31 | \( 1 + (40.1 - 40.1i)T - 961iT^{2} \) |
| 37 | \( 1 + (-19.9 + 37.3i)T + (-760. - 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-24.1 - 16.1i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (38.9 - 31.9i)T + (360. - 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-18.1 - 43.8i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-2.86 + 29.0i)T + (-2.75e3 - 548. i)T^{2} \) |
| 59 | \( 1 + (-58.5 + 17.7i)T + (2.89e3 - 1.93e3i)T^{2} \) |
| 61 | \( 1 + (54.0 + 44.3i)T + (725. + 3.64e3i)T^{2} \) |
| 67 | \( 1 + (25.5 - 31.1i)T + (-875. - 4.40e3i)T^{2} \) |
| 71 | \( 1 + (5.17 + 1.03i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (26.9 - 5.36i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-17.2 + 41.7i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (47.4 - 25.3i)T + (3.82e3 - 5.72e3i)T^{2} \) |
| 89 | \( 1 + (13.9 + 20.9i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-56.7 - 56.7i)T + 9.40e3iT^{2} \) |
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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
−12.35975073141436946488340967199, −11.37729246186137239914692094545, −10.82145648753018241363218051035, −9.453551996109987482293171056888, −8.531196248817126179632229153871, −7.34119621930296233348934145858, −6.81196252832757269100267054282, −4.35177735301449099647183031060, −3.05868563973629620335279531876, −0.30993977356852108207997191791,
2.09434838445391247361613898889, 3.89337721721213707738998648140, 5.83276009392242406694130360071, 7.57772787471436583804438733977, 8.178128748063843993786189183221, 8.789749467828909642982088299312, 10.37844025255988042016869879315, 11.42674960825744963241628015358, 12.16818363577345675004962555721, 13.06876237276417489392194817668
