| 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
−11.83931707254632400398511498361, −10.74428170056822736281693867071, −9.628472202364289058679968012166, −8.765003433579462859653605002208, −7.55815477004274945691091760434, −7.09234744244096257711292104156, −6.05243897934928685968844826960, −3.51697761911856239409649560546, −2.42287865853778680860631788294, −0.02819849338131207927940388931,
3.02718074178915692762075039235, 3.97580839947135665832907209832, 5.84386517241870723871826625822, 6.98627517627947300624947060600, 8.360516652661727887118781016325, 8.879014782636430931960745532302, 9.799936717659815438148230231339, 10.93512406992099861868847648502, 11.35430477967896391967110393195, 12.81166894266972294633951218178
