L(s) = 1 | + (−3.15 − 1.82i)5-s − 1.21·7-s − 3.18i·11-s + (3.24 − 1.87i)13-s + (5.49 + 3.17i)17-s + (−1.51 + 4.08i)19-s + (8.02 − 4.63i)23-s + (4.15 + 7.20i)25-s + (−2.76 − 4.79i)29-s + 7.28i·31-s + (3.84 + 2.22i)35-s − 7.63i·37-s + (5.22 − 9.04i)41-s + (2.58 − 4.47i)43-s + (−9.64 + 5.56i)47-s + ⋯ |
L(s) = 1 | + (−1.41 − 0.815i)5-s − 0.460·7-s − 0.961i·11-s + (0.899 − 0.519i)13-s + (1.33 + 0.769i)17-s + (−0.348 + 0.937i)19-s + (1.67 − 0.965i)23-s + (0.831 + 1.44i)25-s + (−0.513 − 0.889i)29-s + 1.30i·31-s + (0.650 + 0.375i)35-s − 1.25i·37-s + (0.815 − 1.41i)41-s + (0.393 − 0.681i)43-s + (−1.40 + 0.811i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8896714676\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8896714676\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (1.51 - 4.08i)T \) |
good | 5 | \( 1 + (3.15 + 1.82i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 + 3.18iT - 11T^{2} \) |
| 13 | \( 1 + (-3.24 + 1.87i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.49 - 3.17i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-8.02 + 4.63i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.76 + 4.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.28iT - 31T^{2} \) |
| 37 | \( 1 + 7.63iT - 37T^{2} \) |
| 41 | \( 1 + (-5.22 + 9.04i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.58 + 4.47i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (9.64 - 5.56i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.30 + 10.9i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.27 - 7.40i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.41 - 2.45i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.06 - 3.50i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.96 + 3.39i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.32 - 2.29i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.33 + 4.23i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.122iT - 83T^{2} \) |
| 89 | \( 1 + (5.93 + 10.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.33 + 4.23i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.526780902753487310953608100528, −7.939180500835250490991520271359, −7.19405709293101069939052569976, −6.07919348057126287327311310468, −5.51043833470875520849256265452, −4.44213164907589338426131953590, −3.58775097310568135584193734949, −3.19730185950320997709035744642, −1.31131486163877452296161314748, −0.34852035853642008565442169298,
1.25043390292138805069141564276, 2.93772516872523427743082122774, 3.31813940949136584228244146106, 4.33788368209346362499767238523, 5.05478860078453229206092714142, 6.34392390693602034517681863704, 6.93850591072152020974642646303, 7.56295579923598457336531598033, 8.121605320297111371632900058939, 9.306305930285783442206305001933