| L(s) = 1 | + (0.206 − 2.76i)2-s + (−5.60 − 0.844i)4-s + (0.116 + 0.108i)5-s + (−1.85 + 1.88i)7-s + (−2.26 + 9.90i)8-s + (0.322 − 0.299i)10-s + (−1.22 − 0.832i)11-s + (−4.48 + 2.16i)13-s + (4.82 + 5.51i)14-s + (16.0 + 4.95i)16-s + (0.262 − 0.668i)17-s + (−2.79 + 4.83i)19-s + (−0.561 − 0.704i)20-s + (−2.55 + 3.20i)22-s + (−1.20 − 3.07i)23-s + ⋯ |
| L(s) = 1 | + (0.146 − 1.95i)2-s + (−2.80 − 0.422i)4-s + (0.0520 + 0.0483i)5-s + (−0.700 + 0.713i)7-s + (−0.799 + 3.50i)8-s + (0.101 − 0.0946i)10-s + (−0.368 − 0.251i)11-s + (−1.24 + 0.599i)13-s + (1.29 + 1.47i)14-s + (4.01 + 1.23i)16-s + (0.0636 − 0.162i)17-s + (−0.640 + 1.10i)19-s + (−0.125 − 0.157i)20-s + (−0.544 + 0.682i)22-s + (−0.251 − 0.640i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0919258 + 0.0376069i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0919258 + 0.0376069i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.85 - 1.88i)T \) |
| good | 2 | \( 1 + (-0.206 + 2.76i)T + (-1.97 - 0.298i)T^{2} \) |
| 5 | \( 1 + (-0.116 - 0.108i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (1.22 + 0.832i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (4.48 - 2.16i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-0.262 + 0.668i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (2.79 - 4.83i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.20 + 3.07i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (2.33 + 2.93i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-0.368 - 0.639i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.44 - 0.519i)T + (35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (0.880 - 3.85i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-1.46 - 6.40i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-0.771 + 10.3i)T + (-46.4 - 7.00i)T^{2} \) |
| 53 | \( 1 + (10.7 + 1.62i)T + (50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (9.45 - 8.77i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (-3.56 + 0.536i)T + (58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (-3.13 - 5.42i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.474 - 0.594i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.0447 - 0.596i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (-0.0318 + 0.0551i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.67 - 3.69i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.703 - 0.479i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−11.34892217123236746375403952451, −10.23809705599409903806815305246, −9.854757501903867521795828686101, −8.937898743376481359710014246632, −8.056057965438970736231709615610, −6.21697525488532785508368028383, −5.08689219221682617175377272786, −4.07258558200073642139187025229, −2.86772333160482895197599021892, −2.04772743154443820318215570728,
0.05749042338936681716466818422, 3.39129586803919411664770450995, 4.61878403703547638012585168859, 5.40143610872215401639183187608, 6.48898822022742829452501816435, 7.31535499585451450846089210995, 7.79660159257134365013252195754, 9.097441063274862322313745536369, 9.656100803861697776386089493316, 10.67297562939861446876679590466
