L(s) = 1 | + (−0.869 + 1.49i)3-s + 2.66·5-s + (0.767 + 0.442i)7-s + (−1.48 − 2.60i)9-s + (−2.39 + 4.15i)11-s + (−5.90 + 3.40i)13-s + (−2.32 + 3.99i)15-s + (5.55 − 3.20i)17-s + (2.50 + 4.33i)19-s + (−1.33 + 0.764i)21-s + (−1.46 + 0.846i)23-s + 2.12·25-s + (5.19 + 0.0333i)27-s + (5.11 + 2.95i)29-s + (−5.28 − 3.05i)31-s + ⋯ |
L(s) = 1 | + (−0.501 + 0.864i)3-s + 1.19·5-s + (0.289 + 0.167i)7-s + (−0.496 − 0.868i)9-s + (−0.722 + 1.25i)11-s + (−1.63 + 0.944i)13-s + (−0.599 + 1.03i)15-s + (1.34 − 0.778i)17-s + (0.573 + 0.994i)19-s + (−0.290 + 0.166i)21-s + (−0.305 + 0.176i)23-s + 0.425·25-s + (0.999 + 0.00641i)27-s + (0.949 + 0.548i)29-s + (−0.949 − 0.548i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.684834 + 1.12064i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.684834 + 1.12064i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.869 - 1.49i)T \) |
| 67 | \( 1 + (-2.91 + 7.64i)T \) |
good | 5 | \( 1 - 2.66T + 5T^{2} \) |
| 7 | \( 1 + (-0.767 - 0.442i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.39 - 4.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.90 - 3.40i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.55 + 3.20i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.50 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.46 - 0.846i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.11 - 2.95i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.28 + 3.05i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.639 - 1.10i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.45 - 2.51i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 12.1iT - 43T^{2} \) |
| 47 | \( 1 + (-3.58 - 2.06i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 5.25T + 53T^{2} \) |
| 59 | \( 1 + 0.587iT - 59T^{2} \) |
| 61 | \( 1 + (5.85 - 3.38i)T + (30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-1.38 - 0.797i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.71 - 6.44i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.0 - 5.81i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.12 - 4.11i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 9.09iT - 89T^{2} \) |
| 97 | \( 1 + (-7.83 + 4.52i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.08334393947750047575253046916, −9.797050195293373143609574236929, −9.427775919752273183441760069837, −7.907424881133614167553844143853, −7.03750595609253165925267918572, −5.88378022489003738360059439405, −5.13124632818594058763738660694, −4.57571420632529107027157359807, −2.97407507642764100186030293292, −1.81052436658091292735434450138,
0.67814778560607236356638145679, 2.13722143070344701372976821278, 3.08161741941892508536348930791, 5.18017810061075546338207504830, 5.45364249344152728421327131679, 6.35664468804858083755303576697, 7.50043611644381050368925251010, 8.018932123557715086971377949757, 9.147942319858852757591781467442, 10.37194288974927884093027534279