L(s) = 1 | + (−0.841 + 1.13i)2-s + (0.965 + 0.258i)3-s + (−0.583 − 1.91i)4-s + (−0.192 + 0.719i)5-s + (−1.10 + 0.879i)6-s − 3.56·7-s + (2.66 + 0.947i)8-s + (0.866 + 0.499i)9-s + (−0.655 − 0.824i)10-s + (2.95 − 2.95i)11-s + (−0.0683 − 1.99i)12-s + (0.946 − 0.253i)13-s + (3.00 − 4.05i)14-s + (−0.372 + 0.644i)15-s + (−3.31 + 2.23i)16-s + (−2.56 − 4.44i)17-s + ⋯ |
L(s) = 1 | + (−0.595 + 0.803i)2-s + (0.557 + 0.149i)3-s + (−0.291 − 0.956i)4-s + (−0.0861 + 0.321i)5-s + (−0.451 + 0.359i)6-s − 1.34·7-s + (0.942 + 0.334i)8-s + (0.288 + 0.166i)9-s + (−0.207 − 0.260i)10-s + (0.892 − 0.892i)11-s + (−0.0197 − 0.577i)12-s + (0.262 − 0.0703i)13-s + (0.801 − 1.08i)14-s + (−0.0961 + 0.166i)15-s + (−0.829 + 0.558i)16-s + (−0.622 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01338 - 0.149839i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01338 - 0.149839i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 - 1.13i)T \) |
| 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 19 | \( 1 + (1.65 + 4.03i)T \) |
good | 5 | \( 1 + (0.192 - 0.719i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + (-2.95 + 2.95i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.946 + 0.253i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (2.56 + 4.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.06 + 5.30i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (8.89 - 2.38i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 - 1.70T + 31T^{2} \) |
| 37 | \( 1 + (-3.96 + 3.96i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.87 - 3.24i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.50 - 2.54i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-4.03 - 2.33i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.10 + 7.86i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.51 + 13.1i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (0.430 + 1.60i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (2.22 + 8.30i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.97 - 1.14i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.81 + 1.62i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.68 - 2.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.09 - 3.09i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.67 + 9.82i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.01 - 4.62i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.520169609683347163722877054781, −9.258894363167952471489841968792, −8.591723620516167213816132065455, −7.38165617472697681751852208521, −6.70699343375860271355837353180, −6.12634676862615716095188848285, −4.83047152956539974004412403743, −3.64482657313544382547155307228, −2.58891104414562215640841289863, −0.60493212279680191552177727897,
1.34490384317238046821018102248, 2.51091569332640450614810908808, 3.75137717980546885276749741486, 4.18318016419611339916055975667, 5.99200318128466511765879060911, 6.96593474644437465975201649987, 7.72926360989178197930872899937, 8.889025742651949887458751468538, 9.199967889568297696728400251301, 10.00042696708901764691755179596