L(s) = 1 | + (−0.366 + 1.36i)2-s − 1.73·3-s + (−1.73 − i)4-s + (−1.5 − 0.866i)5-s + (0.633 − 2.36i)6-s + (2.59 − 0.5i)7-s + (2 − 1.99i)8-s + 2.99·9-s + (1.73 − 1.73i)10-s + (−0.866 + 0.5i)11-s + (2.99 + 1.73i)12-s + (1.5 + 0.866i)13-s + (−0.267 + 3.73i)14-s + (2.59 + 1.49i)15-s + (1.99 + 3.46i)16-s − 3.46i·17-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s − 1.00·3-s + (−0.866 − 0.5i)4-s + (−0.670 − 0.387i)5-s + (0.258 − 0.965i)6-s + (0.981 − 0.188i)7-s + (0.707 − 0.707i)8-s + 0.999·9-s + (0.547 − 0.547i)10-s + (−0.261 + 0.150i)11-s + (0.866 + 0.499i)12-s + (0.416 + 0.240i)13-s + (−0.0716 + 0.997i)14-s + (0.670 + 0.387i)15-s + (0.499 + 0.866i)16-s − 0.840i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.728137 + 0.134251i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.728137 + 0.134251i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 3 | \( 1 + 1.73T \) |
| 7 | \( 1 + (-2.59 + 0.5i)T \) |
good | 5 | \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + (-4.33 - 2.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.33 + 7.5i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + (-7.5 - 4.33i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.59 + 4.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + (4.33 - 7.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12.9 + 7.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4iT - 71T^{2} \) |
| 73 | \( 1 - 10.3iT - 73T^{2} \) |
| 79 | \( 1 + (-2.59 + 1.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 1.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 17.3iT - 89T^{2} \) |
| 97 | \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.80645776447348793380029164009, −11.34848341954288387488874756433, −10.09959275969914069405439866245, −9.124095635779032269498349203989, −7.76125256263656165143499350037, −7.38877656479686805733736527191, −5.95186984176749184658675205634, −5.00185437173545080128885535718, −4.21545744579474469985517269000, −0.938775135111532964902072001335,
1.29992899800338632119095107146, 3.30143273187698446438651613300, 4.60679245217432409351257916386, 5.55649059205630938416890530759, 7.24939555692379357938558626047, 8.099691198016922819649457921691, 9.277888819617372154354207277328, 10.70033374824078062119403571241, 10.89651310590881379391728429451, 11.83666845495690266113330351057