L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 1.5i)3-s + (0.499 − 0.866i)4-s + (−1.73 − 3i)5-s + 1.73i·6-s + (2 − 1.73i)7-s + 0.999i·8-s + (−1.5 − 2.59i)9-s + (3 + 1.73i)10-s + (−0.866 − 0.5i)11-s + (−0.866 − 1.49i)12-s + 3.46i·13-s + (−0.866 + 2.5i)14-s − 6·15-s + (−0.5 − 0.866i)16-s + (0.866 − 1.5i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.499 − 0.866i)3-s + (0.249 − 0.433i)4-s + (−0.774 − 1.34i)5-s + 0.707i·6-s + (0.755 − 0.654i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.948 + 0.547i)10-s + (−0.261 − 0.150i)11-s + (−0.250 − 0.433i)12-s + 0.960i·13-s + (−0.231 + 0.668i)14-s − 1.54·15-s + (−0.125 − 0.216i)16-s + (0.210 − 0.363i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.393309 - 0.880167i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.393309 - 0.880167i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (1.73 + 3i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (-0.866 + 1.5i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3iT - 29T^{2} \) |
| 31 | \( 1 + (-3 - 1.73i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (-0.866 - 1.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.06 - 10.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9 + 5.19i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8 + 13.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15iT - 71T^{2} \) |
| 73 | \( 1 + (12 + 6.92i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 15.5iT - 97T^{2} \) |
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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
−10.77580621046470573427793974231, −9.373825422312231493155042799556, −8.759328025962668525246036663052, −7.86378415301078788630660370981, −7.54223973649710689887818939125, −6.27782440392599773076516055581, −4.95074603585306449643708346454, −3.90922605867524494959903509466, −1.90284611735174647136568173103, −0.69293433755956247229683301550,
2.39889917273376120677588104461, 3.20259660323436268854874498903, 4.32649131554087306140132486420, 5.68504934958089764262672199406, 7.11811583155427926685247020589, 8.099809327555687157242263061280, 8.482270700801752333784414896482, 9.828203399056617367844100048552, 10.49459781212864453831899850981, 11.11156792867219420626629246180