| L(s) = 1 | + (−1.65 − 1.12i)2-s + (0.729 + 1.85i)4-s + (0.400 − 0.123i)5-s + (−1.90 − 1.83i)7-s + (−0.00157 + 0.00688i)8-s + (−0.800 − 0.246i)10-s + (−0.0436 + 0.582i)11-s + (−3.22 + 1.55i)13-s + (1.08 + 5.17i)14-s + (2.93 − 2.72i)16-s + (0.518 − 0.0780i)17-s + (−0.603 − 1.04i)19-s + (0.521 + 0.653i)20-s + (0.727 − 0.912i)22-s + (−8.89 − 1.34i)23-s + ⋯ |
| L(s) = 1 | + (−1.16 − 0.796i)2-s + (0.364 + 0.929i)4-s + (0.178 − 0.0552i)5-s + (−0.721 − 0.692i)7-s + (−0.000555 + 0.00243i)8-s + (−0.252 − 0.0780i)10-s + (−0.0131 + 0.175i)11-s + (−0.895 + 0.431i)13-s + (0.291 + 1.38i)14-s + (0.734 − 0.681i)16-s + (0.125 − 0.0189i)17-s + (−0.138 − 0.239i)19-s + (0.116 + 0.146i)20-s + (0.155 − 0.194i)22-s + (−1.85 − 0.279i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0407928 + 0.0539624i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0407928 + 0.0539624i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (1.90 + 1.83i)T \) |
| good | 2 | \( 1 + (1.65 + 1.12i)T + (0.730 + 1.86i)T^{2} \) |
| 5 | \( 1 + (-0.400 + 0.123i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.0436 - 0.582i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (3.22 - 1.55i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-0.518 + 0.0780i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (0.603 + 1.04i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (8.89 + 1.34i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-2.11 - 2.65i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (3.57 - 6.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.29 + 3.29i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (0.789 - 3.46i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-1.23 - 5.42i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-4.74 - 3.23i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (-1.18 - 3.01i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (8.24 + 2.54i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-2.07 + 5.29i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (3.72 - 6.45i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.36 + 6.73i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (10.4 - 7.12i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (4.12 + 7.14i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (15.5 + 7.50i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.281 - 3.75i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 5.47T + 97T^{2} \) |
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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
−11.13027119869077495374311859568, −10.24174301941274007087706736027, −9.754769411115985459697468780845, −9.005451434944436355570860703757, −7.88424861906387684694681281493, −7.08676477251406891132084186926, −5.83100466942981813743533400020, −4.35873151820070992379439406105, −2.98608990873516531330219551150, −1.70267170836675000924792545222,
0.05915358851979016346550314527, 2.28872784190458374195367728490, 3.87047994147145885964781393641, 5.67317477153855025387867136464, 6.21842423206846039860943026425, 7.38375804838769263038608075050, 8.104732312069757200911790052901, 9.004548302500889598266725224326, 9.914160531376694296021164021521, 10.20905685449980949659255055373
