L(s) = 1 | + (1.38 + 2.39i)2-s + (0.5 + 0.866i)3-s + (−2.83 + 4.91i)4-s + 3.07·5-s + (−1.38 + 2.39i)6-s + (1.16 − 2.02i)7-s − 10.1·8-s + (−0.499 + 0.866i)9-s + (4.26 + 7.37i)10-s + (0.5 + 0.866i)11-s − 5.66·12-s + (−3.02 − 1.95i)13-s + 6.46·14-s + (1.53 + 2.66i)15-s + (−8.40 − 14.5i)16-s + (−0.0407 + 0.0705i)17-s + ⋯ |
L(s) = 1 | + (0.979 + 1.69i)2-s + (0.288 + 0.499i)3-s + (−1.41 + 2.45i)4-s + 1.37·5-s + (−0.565 + 0.979i)6-s + (0.441 − 0.764i)7-s − 3.59·8-s + (−0.166 + 0.288i)9-s + (1.34 + 2.33i)10-s + (0.150 + 0.261i)11-s − 1.63·12-s + (−0.840 − 0.542i)13-s + 1.72·14-s + (0.397 + 0.687i)15-s + (−2.10 − 3.63i)16-s + (−0.00988 + 0.0171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.406289 + 2.64356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.406289 + 2.64356i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (3.02 + 1.95i)T \) |
good | 2 | \( 1 + (-1.38 - 2.39i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 7 | \( 1 + (-1.16 + 2.02i)T + (-3.5 - 6.06i)T^{2} \) |
| 17 | \( 1 + (0.0407 - 0.0705i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.32 + 5.75i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.19 - 2.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.746 - 1.29i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 + (3.23 + 5.60i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.47 + 7.74i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.82 - 3.15i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + (4.09 - 7.08i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.94 - 10.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.50 - 4.33i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 9.99T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 0.656T + 83T^{2} \) |
| 89 | \( 1 + (4.64 + 8.04i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.373 - 0.646i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.89112557111736758646498833287, −10.43318409988004756376981537543, −9.414792027943456516142447818371, −8.755810446459410971128404038388, −7.43412812033798803845153639812, −7.00252330678476038887870313714, −5.61874168774453845146139980722, −5.15131749881976017322216139071, −4.12940292342387150303370093045, −2.78470158993158639501449318662,
1.58361589656972876604452661157, 2.23528958987429076405497817680, 3.34283938208491228895629729439, 4.89247784711249293122099600803, 5.60768602173260271553437267931, 6.48907224750077394763542501966, 8.450917175647153608412216721997, 9.475068274640019672199403432591, 9.887351755315796605854688508519, 10.92594335181507900797542449961