L(s) = 1 | − 2.58i·2-s + (0.259 + 0.449i)3-s − 4.70·4-s + (1.39 − 0.806i)5-s + (1.16 − 0.671i)6-s + 6.99i·8-s + (1.36 − 2.36i)9-s + (−2.08 − 3.61i)10-s + (2.34 − 1.35i)11-s + (−1.21 − 2.11i)12-s + (−2.36 − 2.71i)13-s + (0.723 + 0.417i)15-s + 8.69·16-s + 3.12·17-s + (−6.12 − 3.53i)18-s + (−3.18 − 1.84i)19-s + ⋯ |
L(s) = 1 | − 1.83i·2-s + (0.149 + 0.259i)3-s − 2.35·4-s + (0.624 − 0.360i)5-s + (0.474 − 0.273i)6-s + 2.47i·8-s + (0.455 − 0.788i)9-s + (−0.659 − 1.14i)10-s + (0.706 − 0.407i)11-s + (−0.351 − 0.609i)12-s + (−0.656 − 0.753i)13-s + (0.186 + 0.107i)15-s + 2.17·16-s + 0.758·17-s + (−1.44 − 0.833i)18-s + (−0.731 − 0.422i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.133519 + 1.39418i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.133519 + 1.39418i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.36 + 2.71i)T \) |
good | 2 | \( 1 + 2.58iT - 2T^{2} \) |
| 3 | \( 1 + (-0.259 - 0.449i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.39 + 0.806i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.34 + 1.35i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 + (3.18 + 1.84i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 1.98T + 23T^{2} \) |
| 29 | \( 1 + (-2.68 + 4.65i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (9.07 + 5.23i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.95iT - 37T^{2} \) |
| 41 | \( 1 + (-6.66 - 3.85i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.67 + 2.90i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.913 - 0.527i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.63 - 6.29i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 11.4iT - 59T^{2} \) |
| 61 | \( 1 + (1.46 - 2.53i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.7 + 6.79i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.17 + 0.675i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.88 - 4.55i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.10 - 5.37i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.69iT - 83T^{2} \) |
| 89 | \( 1 + 1.75iT - 89T^{2} \) |
| 97 | \( 1 + (-13.4 + 7.74i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.00185559597377109650137157290, −9.598481166752857452531609813536, −8.963925331897294553324677857975, −7.87324272189694621114847030001, −6.24668674200776192403950545026, −5.13077070349787715490189604508, −4.08923989571851463249925153024, −3.27469693279845228844432075586, −2.05303987804908237864856313611, −0.793493679917535543387455334819,
1.94211240745997086763732667232, 3.96050445208946493191239997078, 4.93476340073072764356582877754, 5.84219715830201404193525699508, 6.78198900679245237612170341609, 7.27080813444731917476507422068, 8.151477962192414011172463059145, 9.098793352743698306682796923932, 9.805621896318692973851634948830, 10.69483182345599992039326196523