L(s) = 1 | + 1.73i·3-s + (−0.5 − 0.866i)5-s + (1.5 − 2.59i)7-s − 2.99·9-s + (2.5 − 4.33i)11-s + (−2.5 − 4.33i)13-s + (1.49 − 0.866i)15-s − 2·17-s + 4·19-s + (4.5 + 2.59i)21-s + (0.5 + 0.866i)23-s + (2 − 3.46i)25-s − 5.19i·27-s + (−4.5 + 7.79i)29-s + (0.5 + 0.866i)31-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + (−0.223 − 0.387i)5-s + (0.566 − 0.981i)7-s − 0.999·9-s + (0.753 − 1.30i)11-s + (−0.693 − 1.20i)13-s + (0.387 − 0.223i)15-s − 0.485·17-s + 0.917·19-s + (0.981 + 0.566i)21-s + (0.104 + 0.180i)23-s + (0.400 − 0.692i)25-s − 0.999i·27-s + (−0.835 + 1.44i)29-s + (0.0898 + 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25382 - 0.456355i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25382 - 0.456355i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.72893673782673732821568918200, −9.819470853695833479821261566070, −8.862585617961254780740691537800, −8.166221704813439176829033591306, −7.15647178722108643884658545671, −5.76001652068867621954264549495, −4.95399701410627557008623306328, −3.96432099671341392538325693975, −3.06682822576520482311985729038, −0.796048099372776484310285925970,
1.73617124899031196051061078570, 2.56771223640190494402709759702, 4.25896136730717251019945064395, 5.36825962588137218970405058696, 6.53189690748097713866512049406, 7.18959979732730806452030239329, 7.987446161744853082248826737896, 9.130423303298720833791456955605, 9.655315308619979168593484329107, 11.35189272500280184237064208959