| L(s) = 1 | + (−1.52 + 1.04i)2-s + (0.514 − 1.31i)4-s + (1.32 + 0.410i)5-s + (−1.73 − 1.99i)7-s + (−0.242 − 1.06i)8-s + (−2.45 + 0.757i)10-s + (0.215 + 2.87i)11-s + (4.03 + 1.94i)13-s + (4.72 + 1.23i)14-s + (3.54 + 3.28i)16-s + (0.476 + 0.0718i)17-s + (1.90 − 3.30i)19-s + (1.22 − 1.53i)20-s + (−3.31 − 4.16i)22-s + (6.73 − 1.01i)23-s + ⋯ |
| L(s) = 1 | + (−1.07 + 0.735i)2-s + (0.257 − 0.655i)4-s + (0.594 + 0.183i)5-s + (−0.656 − 0.754i)7-s + (−0.0859 − 0.376i)8-s + (−0.776 + 0.239i)10-s + (0.0649 + 0.866i)11-s + (1.12 + 0.539i)13-s + (1.26 + 0.330i)14-s + (0.885 + 0.821i)16-s + (0.115 + 0.0174i)17-s + (0.437 − 0.757i)19-s + (0.273 − 0.342i)20-s + (−0.707 − 0.887i)22-s + (1.40 − 0.211i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.695815 + 0.474992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.695815 + 0.474992i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.73 + 1.99i)T \) |
| good | 2 | \( 1 + (1.52 - 1.04i)T + (0.730 - 1.86i)T^{2} \) |
| 5 | \( 1 + (-1.32 - 0.410i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.215 - 2.87i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (-4.03 - 1.94i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-0.476 - 0.0718i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-1.90 + 3.30i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.73 + 1.01i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (1.45 - 1.81i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-3.94 - 6.82i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.50 - 8.92i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-1.56 - 6.84i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.546 + 2.39i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-9.25 + 6.30i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (1.96 - 4.99i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (3.57 - 1.10i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-0.0147 - 0.0376i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (0.534 + 0.926i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.21 + 5.28i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (6.78 + 4.62i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-6.91 + 11.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.465 + 0.224i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.0586 - 0.782i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
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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
−10.84656343725082568649842704876, −10.13657707096604265330351989484, −9.376306137996913114730353109660, −8.716929211879700654120856621069, −7.50565247837047245445902413494, −6.79356689576163702282258666406, −6.17335516970631950254977896737, −4.57027411136076427604850977234, −3.19345675901726979667407833394, −1.20492727604732733487362373073,
0.967917335077878519646490341673, 2.43956809169151425121809480243, 3.53664357355001658535777622051, 5.63071110975940362927116049997, 5.94327646676321701183817557471, 7.64501496480909435322036197855, 8.633283420618070922424868070098, 9.234829177527490896028373904907, 9.898543969461108377862942029344, 10.90912303331941113761791097094
