| L(s) = 1 | + (−0.573 − 0.819i)2-s + (−0.133 − 1.52i)3-s + (−0.342 + 0.939i)4-s + (−1.17 + 0.984i)6-s + (−1.06 + 3.99i)7-s + (0.965 − 0.258i)8-s + (0.642 − 0.113i)9-s + (1.83 − 3.17i)11-s + (1.47 + 0.396i)12-s + (−6.20 − 0.542i)13-s + (3.88 − 1.41i)14-s + (−0.766 − 0.642i)16-s + (1.86 − 1.30i)17-s + (−0.461 − 0.461i)18-s + (−4.31 + 0.639i)19-s + ⋯ |
| L(s) = 1 | + (−0.405 − 0.579i)2-s + (−0.0770 − 0.881i)3-s + (−0.171 + 0.469i)4-s + (−0.479 + 0.402i)6-s + (−0.404 + 1.50i)7-s + (0.341 − 0.0915i)8-s + (0.214 − 0.0377i)9-s + (0.551 − 0.956i)11-s + (0.427 + 0.114i)12-s + (−1.72 − 0.150i)13-s + (1.03 − 0.377i)14-s + (−0.191 − 0.160i)16-s + (0.451 − 0.316i)17-s + (−0.108 − 0.108i)18-s + (−0.989 + 0.146i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0462953 + 0.0950359i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0462953 + 0.0950359i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.573 + 0.819i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (4.31 - 0.639i)T \) |
| good | 3 | \( 1 + (0.133 + 1.52i)T + (-2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (1.06 - 3.99i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.83 + 3.17i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (6.20 + 0.542i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.86 + 1.30i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (7.21 - 3.36i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (1.32 + 7.52i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (5.27 - 3.04i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.80 - 5.80i)T - 37iT^{2} \) |
| 41 | \( 1 + (3.86 - 4.60i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.23 + 2.65i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-3.35 + 4.79i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (4.12 + 8.85i)T + (-34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (1.07 - 6.09i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.57 - 0.938i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-9.13 - 6.39i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-2.18 - 6.01i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-3.47 + 0.304i)T + (71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (6.41 + 5.37i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-9.20 - 2.46i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.289 - 0.242i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (8.25 + 11.7i)T + (-33.1 + 91.1i)T^{2} \) |
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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
−9.661726434941948547312355977375, −8.632445314218043401999919089822, −7.997530866100528188923066404627, −7.01331592627472130365742913683, −6.13117457966272573150394027266, −5.27041254640958410082095040801, −3.80980239602241746934648516434, −2.58584314359838736628734920970, −1.83458914742486572510776713666, −0.05353007767846279711636892618,
1.89034081537927764441319991421, 3.83045083341342409412569918112, 4.35385100523441787944454230671, 5.21767770599868625828151873364, 6.62804803384086877474502801220, 7.19210978436250385176421025598, 7.85309997618740581668048979630, 9.273064524647584851417019971431, 9.680258876838944006984711976691, 10.47712705750900011611011730421
