L(s) = 1 | + (1.18 + 1.26i)3-s + (0.686 − 0.396i)5-s + 2.37·7-s + (−0.186 + 2.99i)9-s + 3.46i·11-s + (−2.87 − 1.65i)13-s + (1.31 + 0.396i)15-s + (0.686 − 0.396i)17-s + (4 − 1.73i)19-s + (2.81 + 2.99i)21-s + (6.43 + 3.71i)23-s + (−2.18 + 3.78i)25-s + (−4.00 + 3.31i)27-s + (−2.68 + 4.65i)29-s − 4.40i·31-s + ⋯ |
L(s) = 1 | + (0.684 + 0.728i)3-s + (0.306 − 0.177i)5-s + 0.896·7-s + (−0.0620 + 0.998i)9-s + 1.04i·11-s + (−0.796 − 0.459i)13-s + (0.339 + 0.102i)15-s + (0.166 − 0.0960i)17-s + (0.917 − 0.397i)19-s + (0.614 + 0.653i)21-s + (1.34 + 0.774i)23-s + (−0.437 + 0.757i)25-s + (−0.769 + 0.638i)27-s + (−0.498 + 0.863i)29-s − 0.790i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.94082 + 1.14381i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.94082 + 1.14381i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.18 - 1.26i)T \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 5 | \( 1 + (-0.686 + 0.396i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + (2.87 + 1.65i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.686 + 0.396i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-6.43 - 3.71i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.68 - 4.65i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.40iT - 31T^{2} \) |
| 37 | \( 1 + 7.86iT - 37T^{2} \) |
| 41 | \( 1 + (0.313 + 0.543i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.127 - 0.221i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.68 - 2.12i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.68 + 9.84i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.68 - 6.38i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.2 + 5.91i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.31 - 5.73i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.87 - 4.97i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.98 - 5.76i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (-3.68 + 6.38i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.0 + 6.38i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.00680024252806936716576032271, −9.439346058757288646143455451382, −8.768281478334844893487895876157, −7.47473533498303976394752899483, −7.39215106665300541929020180618, −5.35027901591189973234906641610, −5.10026514747757044155259599344, −3.97109712721241513517429721697, −2.78260059685038938493390978920, −1.68142902616280834193807810677,
1.11150631273840534052389915597, 2.34350109767437656404990814574, 3.29560626665196015554460041421, 4.60808663376182853960928729684, 5.70634139517173963621952683379, 6.64496481327307229168511369872, 7.51752990345676080537005093523, 8.245693704330728005484107264758, 8.945965418591055021936772507753, 9.832849306492092429710402171332