| L(s) = 1 | + (0.669 − 0.743i)2-s + (−0.913 + 0.406i)3-s + (−0.104 − 0.994i)4-s + (3.65 − 0.777i)5-s + (−0.309 + 0.951i)6-s + (2.31 + 1.27i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (1.87 − 3.23i)10-s + (−3.31 − 0.0414i)11-s + (0.5 + 0.866i)12-s + (1.49 + 4.60i)13-s + (2.49 − 0.871i)14-s + (−3.02 + 2.19i)15-s + (−0.978 + 0.207i)16-s + (1.21 + 1.34i)17-s + ⋯ |
| L(s) = 1 | + (0.473 − 0.525i)2-s + (−0.527 + 0.234i)3-s + (−0.0522 − 0.497i)4-s + (1.63 − 0.347i)5-s + (−0.126 + 0.388i)6-s + (0.876 + 0.481i)7-s + (−0.286 − 0.207i)8-s + (0.223 − 0.247i)9-s + (0.591 − 1.02i)10-s + (−0.999 − 0.0124i)11-s + (0.144 + 0.249i)12-s + (0.415 + 1.27i)13-s + (0.667 − 0.232i)14-s + (−0.781 + 0.567i)15-s + (−0.244 + 0.0519i)16-s + (0.294 + 0.326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94721 - 0.627796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94721 - 0.627796i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.669 + 0.743i)T \) |
| 3 | \( 1 + (0.913 - 0.406i)T \) |
| 7 | \( 1 + (-2.31 - 1.27i)T \) |
| 11 | \( 1 + (3.31 + 0.0414i)T \) |
| good | 5 | \( 1 + (-3.65 + 0.777i)T + (4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (-1.49 - 4.60i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.21 - 1.34i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.679 + 6.46i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (0.282 + 0.488i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.00462 - 0.00335i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (5.81 + 1.23i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (-5.65 - 2.51i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (5.08 + 3.69i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 1.95T + 43T^{2} \) |
| 47 | \( 1 + (-1.12 + 10.7i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (3.84 + 0.816i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (0.289 + 2.75i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (11.7 - 2.48i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (7.68 - 13.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.0939 + 0.289i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.29 - 12.3i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (11.4 - 12.6i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.625 + 1.92i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (2.80 + 4.86i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.80 + 11.7i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.02754890223996440833204457612, −10.20296562513089401245039718504, −9.336226436150478814877703791214, −8.633215945934825841751133588889, −6.96307623343105407219184677989, −5.83669703630956184452196601507, −5.27780174493203101562377888128, −4.43272463841266536502378016127, −2.54164431260310802728485648210, −1.57323814023828742929069879007,
1.63074917182188069059151131942, 3.08694793128225455931625258627, 4.81700926246719047090192441974, 5.67639598112167804751908314054, 6.08165482802718807708790225990, 7.48761846392926817015490199523, 8.023987774853954422067438085879, 9.483625179448682594554421876476, 10.50825636698942890270242332398, 10.83687469001388728721285738862
