| L(s) = 1 | + (−0.766 − 0.642i)2-s + (1.20 − 0.439i)3-s + (0.173 + 0.984i)4-s + (0.641 − 3.63i)5-s + (−1.20 − 0.439i)6-s + (0.221 + 0.383i)7-s + (0.500 − 0.866i)8-s + (−1.03 + 0.868i)9-s + (−2.83 + 2.37i)10-s + (2.01 − 3.48i)11-s + (0.642 + 1.11i)12-s + (4.59 + 1.67i)13-s + (0.0768 − 0.435i)14-s + (−0.824 − 4.67i)15-s + (−0.939 + 0.342i)16-s + (0.204 + 0.171i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.696 − 0.253i)3-s + (0.0868 + 0.492i)4-s + (0.287 − 1.62i)5-s + (−0.492 − 0.179i)6-s + (0.0836 + 0.144i)7-s + (0.176 − 0.306i)8-s + (−0.345 + 0.289i)9-s + (−0.895 + 0.751i)10-s + (0.607 − 1.05i)11-s + (0.185 + 0.321i)12-s + (1.27 + 0.464i)13-s + (0.0205 − 0.116i)14-s + (−0.212 − 1.20i)15-s + (−0.234 + 0.0855i)16-s + (0.0495 + 0.0416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.828785 - 1.28541i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.828785 - 1.28541i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-1.20 + 0.439i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.641 + 3.63i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.221 - 0.383i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.01 + 3.48i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.59 - 1.67i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.204 - 0.171i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (1.59 + 9.06i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.171 + 0.143i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.73 - 3.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.44T + 37T^{2} \) |
| 41 | \( 1 + (7.39 - 2.68i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.978 - 5.54i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.68 + 1.41i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (1.72 + 9.79i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-2.68 - 2.25i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.708 - 4.01i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.113 - 0.0949i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.99 - 11.2i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.33 - 0.486i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-9.58 + 3.48i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.64 - 2.84i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.60 + 2.03i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (3.12 + 2.62i)T + (16.8 + 95.5i)T^{2} \) |
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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
−9.938148736282664661767610879972, −8.882699868886773680772296516719, −8.496911794573822705009495959094, −8.297382402993849715542764164070, −6.64684736147663940122375156463, −5.64002013975080306652370339894, −4.47298719799706378045317386260, −3.37887009829581101670166761947, −1.99256572212275942211374448471, −0.934199539526574832160565747834,
1.83211250542855158448180827818, 3.12868678221070411342663969462, 3.91439798168054838835381105791, 5.68001307338622193173503795964, 6.43402274579687189358567986890, 7.26817530202576514137652745739, 7.999791548602563673573372824555, 9.092768146388730555509671977540, 9.717178480122506359675573032216, 10.47675950587783664658653480062
