L(s) = 1 | + (0.5 + 0.866i)2-s + (0.258 − 0.448i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 1.5i)5-s + 0.517·6-s + (−0.965 − 0.258i)7-s − 0.999·8-s + (0.366 + 0.633i)9-s + (0.866 − 1.5i)10-s + (0.965 − 1.67i)11-s + (0.258 + 0.448i)12-s + (−0.258 − 0.965i)14-s − 0.896·15-s + (−0.5 − 0.866i)16-s + (−0.366 + 0.633i)18-s + (−0.965 − 1.67i)19-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (0.258 − 0.448i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 1.5i)5-s + 0.517·6-s + (−0.965 − 0.258i)7-s − 0.999·8-s + (0.366 + 0.633i)9-s + (0.866 − 1.5i)10-s + (0.965 − 1.67i)11-s + (0.258 + 0.448i)12-s + (−0.258 − 0.965i)14-s − 0.896·15-s + (−0.5 − 0.866i)16-s + (−0.366 + 0.633i)18-s + (−0.965 − 1.67i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.001140863\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001140863\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.965 + 0.258i)T \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 0.517T + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.281715104094288726791718873663, −8.868998233162937466676745851911, −8.238615381514168088135348767588, −7.41363987752470024825899005507, −6.58157952463568203381390544560, −5.73421378255363673077359962603, −4.60471448050339605190701809929, −4.03576882114875450705771651254, −2.94684129174437532275751933013, −0.76277230857141769827968820270,
2.03759153083777961913956844131, 3.15870938995719671389297045974, 3.89126020310432153426250619893, 4.31470339002634743473703539665, 6.06081160318513268323585271249, 6.64380353647122783945187504257, 7.43007179956295719388764162838, 8.828707158511858777392123242960, 9.675792138844511465544592462585, 10.17268815689527154751327040817