L(s) = 1 | + (−0.439 − 0.761i)2-s + (0.613 − 1.06i)4-s + 1.34·5-s − 2.83·8-s + (−0.592 − 1.02i)10-s − 1.65·11-s + (−1.68 − 2.91i)13-s + (0.0209 + 0.0362i)16-s + (−0.233 − 0.405i)17-s + (−1.61 + 2.79i)19-s + (0.826 − 1.43i)20-s + (0.726 + 1.25i)22-s − 8.94·23-s − 3.18·25-s + (−1.48 + 2.56i)26-s + ⋯ |
L(s) = 1 | + (−0.310 − 0.538i)2-s + (0.306 − 0.531i)4-s + 0.602·5-s − 1.00·8-s + (−0.187 − 0.324i)10-s − 0.498·11-s + (−0.467 − 0.809i)13-s + (0.00523 + 0.00906i)16-s + (−0.0567 − 0.0982i)17-s + (−0.370 + 0.641i)19-s + (0.184 − 0.320i)20-s + (0.154 + 0.268i)22-s − 1.86·23-s − 0.636·25-s + (−0.290 + 0.503i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8126010634\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8126010634\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.439 + 0.761i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 13 | \( 1 + (1.68 + 2.91i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.233 + 0.405i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.61 - 2.79i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 8.94T + 23T^{2} \) |
| 29 | \( 1 + (-3.13 + 5.42i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.61 + 7.99i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.61 - 7.99i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.70 + 2.95i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.20 + 3.82i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.67 + 8.10i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.286 + 0.497i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.19 + 9.00i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.81 - 6.61i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.298 - 0.516i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.554T + 71T^{2} \) |
| 73 | \( 1 + (-1.02 - 1.77i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.20 - 2.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.52 + 13.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.54 - 7.86i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.949 - 1.64i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.629867794742222615189585580860, −8.404167919143817842450558107210, −7.81197770432902797217795350750, −6.51515796867153071927588266241, −5.90601985289659748380657007779, −5.17303071797094609868885038061, −3.84770615001779251475183546967, −2.54897477651709362059081427405, −1.91511819307399926478080227636, −0.33015190191875693820466365416,
1.92621702527639761417071294844, 2.84734102705517232696322248630, 4.06113187401531617900241813433, 5.15366061712346839180186610648, 6.18551315721138202722753578210, 6.74887630620741640569517526877, 7.62510488193513995428064870271, 8.377311333009546248999868837547, 9.112013318945019001271053970614, 9.897525593215683362785621862877