| L(s) = 1 | + (−1.41 + 0.103i)2-s + (0.584 + 1.63i)3-s + (1.97 − 0.291i)4-s + (−2.25 − 0.819i)5-s + (−0.992 − 2.23i)6-s + (−4.05 − 0.715i)7-s + (−2.76 + 0.615i)8-s + (−2.31 + 1.90i)9-s + (3.25 + 0.923i)10-s + (−1.59 − 4.39i)11-s + (1.63 + 3.05i)12-s + (2.03 + 2.42i)13-s + (5.79 + 0.589i)14-s + (0.0206 − 4.14i)15-s + (3.83 − 1.15i)16-s + (−5.30 + 3.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 + 0.0730i)2-s + (0.337 + 0.941i)3-s + (0.989 − 0.145i)4-s + (−1.00 − 0.366i)5-s + (−0.405 − 0.914i)6-s + (−1.53 − 0.270i)7-s + (−0.976 + 0.217i)8-s + (−0.772 + 0.635i)9-s + (1.03 + 0.291i)10-s + (−0.482 − 1.32i)11-s + (0.470 + 0.882i)12-s + (0.564 + 0.672i)13-s + (1.54 + 0.157i)14-s + (0.00533 − 1.07i)15-s + (0.957 − 0.288i)16-s + (−1.28 + 0.743i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00679002 - 0.0268552i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00679002 - 0.0268552i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.103i)T \) |
| 3 | \( 1 + (-0.584 - 1.63i)T \) |
| good | 5 | \( 1 + (2.25 + 0.819i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (4.05 + 0.715i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.59 + 4.39i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.03 - 2.42i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (5.30 - 3.06i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.49 + 2.58i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.264 - 1.49i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.88 - 1.58i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (8.06 - 1.42i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.515 + 0.297i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.89 - 5.82i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (4.63 - 1.68i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.750 + 4.25i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 + (0.703 - 1.93i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (7.44 + 1.31i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.64 + 4.73i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.78 - 3.08i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.85 - 10.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.66 - 7.94i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.79 - 6.90i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (7.85 + 4.53i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.05 - 1.47i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.81166894266972294633951218178, −11.35430477967896391967110393195, −10.93512406992099861868847648502, −9.799936717659815438148230231339, −8.879014782636430931960745532302, −8.360516652661727887118781016325, −6.98627517627947300624947060600, −5.84386517241870723871826625822, −3.97580839947135665832907209832, −3.02718074178915692762075039235,
0.02819849338131207927940388931, 2.42287865853778680860631788294, 3.51697761911856239409649560546, 6.05243897934928685968844826960, 7.09234744244096257711292104156, 7.55815477004274945691091760434, 8.765003433579462859653605002208, 9.628472202364289058679968012166, 10.74428170056822736281693867071, 11.83931707254632400398511498361
