| L(s) = 1 | + (0.0752 + 0.125i)2-s + (−2.94 + 0.555i)3-s + (1.86 − 3.51i)4-s + (0.165 + 1.52i)5-s + (−0.291 − 0.327i)6-s + (−1.36 + 0.458i)7-s + (1.16 − 0.0630i)8-s + (8.38 − 3.27i)9-s + (−0.178 + 0.135i)10-s + (−9.09 − 6.16i)11-s + (−3.54 + 11.3i)12-s + (−10.4 − 9.90i)13-s + (−0.159 − 0.135i)14-s + (−1.33 − 4.39i)15-s + (−8.83 − 13.0i)16-s + (10.3 − 30.7i)17-s + ⋯ |
| L(s) = 1 | + (0.0376 + 0.0625i)2-s + (−0.982 + 0.185i)3-s + (0.465 − 0.878i)4-s + (0.0331 + 0.304i)5-s + (−0.0485 − 0.0545i)6-s + (−0.194 + 0.0655i)7-s + (0.145 − 0.00788i)8-s + (0.931 − 0.363i)9-s + (−0.0178 + 0.0135i)10-s + (−0.827 − 0.560i)11-s + (−0.295 + 0.949i)12-s + (−0.804 − 0.761i)13-s + (−0.0114 − 0.00970i)14-s + (−0.0889 − 0.293i)15-s + (−0.552 − 0.814i)16-s + (0.610 − 1.81i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.533229 - 0.686377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.533229 - 0.686377i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.94 - 0.555i)T \) |
| 59 | \( 1 + (-58.6 - 6.42i)T \) |
| good | 2 | \( 1 + (-0.0752 - 0.125i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (-0.165 - 1.52i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (1.36 - 0.458i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (9.09 + 6.16i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (10.4 + 9.90i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (-10.3 + 30.7i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (-0.113 - 0.692i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (20.7 - 5.75i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (-1.90 + 3.17i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (-6.31 + 38.5i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (2.40 - 44.3i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (41.4 + 11.5i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (-32.1 - 47.3i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (1.01 - 9.31i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (-18.1 + 23.8i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (-81.1 + 48.7i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (-5.64 - 104. i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (-8.13 + 74.7i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (0.598 - 0.704i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (5.75 - 14.4i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (-31.4 - 67.9i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (-53.3 + 88.6i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (22.6 + 26.6i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−11.87633266276417476748662416821, −11.21151675308461761024258228725, −10.13532487070813894207427376463, −9.741679929926541673294606974225, −7.74241315967680339374401602547, −6.71837193732020039068663303623, −5.63985105977439382140368125270, −4.92781601534116323422894085212, −2.77297804958337207221396769684, −0.55994898770589087817654466525,
2.01130749788612062325092164676, 3.94590177507338320503078723747, 5.20078223732219503157913384654, 6.56033986086351499355963670858, 7.41805067174617529201431932792, 8.493349540998153520091975459500, 10.09267689664559076034455446966, 10.80200044436738918160335717818, 12.11682904967945463777773483271, 12.42868206446020331577777747905
