| L(s) = 1 | + (1.03 + 1.38i)3-s + (1.93 + 0.705i)5-s + (0.744 + 0.131i)7-s + (−0.856 + 2.87i)9-s + (0.949 + 2.60i)11-s + (0.421 + 0.502i)13-s + (1.02 + 3.41i)15-s + (−3.79 + 2.18i)17-s + (−0.155 + 0.270i)19-s + (0.588 + 1.16i)21-s + (−1.32 − 7.53i)23-s + (−0.575 − 0.482i)25-s + (−4.87 + 1.78i)27-s + (5.89 + 4.94i)29-s + (4.16 − 0.733i)31-s + ⋯ |
| L(s) = 1 | + (0.597 + 0.801i)3-s + (0.866 + 0.315i)5-s + (0.281 + 0.0496i)7-s + (−0.285 + 0.958i)9-s + (0.286 + 0.786i)11-s + (0.116 + 0.139i)13-s + (0.265 + 0.882i)15-s + (−0.919 + 0.531i)17-s + (−0.0357 + 0.0619i)19-s + (0.128 + 0.255i)21-s + (−0.276 − 1.57i)23-s + (−0.115 − 0.0964i)25-s + (−0.938 + 0.344i)27-s + (1.09 + 0.918i)29-s + (0.747 − 0.131i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64906 + 1.46659i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64906 + 1.46659i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.03 - 1.38i)T \) |
| good | 5 | \( 1 + (-1.93 - 0.705i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.744 - 0.131i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.949 - 2.60i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.421 - 0.502i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.79 - 2.18i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.155 - 0.270i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.32 + 7.53i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.89 - 4.94i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.16 + 0.733i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.70 + 1.56i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.31 - 5.14i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.00 + 1.09i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.89 + 10.7i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.876T + 53T^{2} \) |
| 59 | \( 1 + (2.94 - 8.09i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (12.1 + 2.14i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (3.32 - 2.79i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-7.08 - 12.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.17 + 12.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.09 - 8.44i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.10 + 8.47i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.55 - 3.20i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.59 + 0.579i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.34388486532983068873492017034, −9.563878758180801314239220039394, −8.781279777226522645407602434935, −8.074331553622457466038729672985, −6.82220879991566001318630731854, −6.05144638910827301907791992333, −4.79034992230689340620659434221, −4.19357228748717677284972858901, −2.75770479556219232430300350172, −1.93245401195563654905393976748,
1.06527621041691641599017681097, 2.19878733772643247844807833641, 3.27589446794209725117333494731, 4.60296949169478677062483224929, 5.88342817125886279193915312669, 6.37813240987995313678408503676, 7.56314439385086830542028168159, 8.232074899434281281603214326572, 9.232972625104214030194479585863, 9.569237690885581163289615436320
