L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.499i)6-s + (−3.76 + 2.17i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−2.04 − 1.17i)11-s + 0.999·12-s + (1.69 − 3.18i)13-s − 4.34·14-s + (−0.5 + 0.866i)16-s + (−1.50 − 2.60i)17-s − 0.999i·18-s + (−0.585 + 0.338i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (0.353 − 0.204i)6-s + (−1.42 + 0.821i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.615 − 0.355i)11-s + 0.288·12-s + (0.469 − 0.883i)13-s − 1.16·14-s + (−0.125 + 0.216i)16-s + (−0.364 − 0.631i)17-s − 0.235i·18-s + (−0.134 + 0.0776i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9740597210\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9740597210\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1.69 + 3.18i)T \) |
good | 7 | \( 1 + (3.76 - 2.17i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.04 + 1.17i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.50 + 2.60i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.585 - 0.338i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.22 + 5.58i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.82 - 8.35i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.11iT - 31T^{2} \) |
| 37 | \( 1 + (6.48 + 3.74i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.60 + 1.50i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.41 + 5.91i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.61iT - 47T^{2} \) |
| 53 | \( 1 + 9.43T + 53T^{2} \) |
| 59 | \( 1 + (-4.56 + 2.63i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.15 - 3.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.04 - 2.91i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.52 + 1.45i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 7.67iT - 73T^{2} \) |
| 79 | \( 1 - 3.74T + 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 + (4.15 + 2.39i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.1 - 8.17i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.775595472107849290409648308866, −8.183662948297653733631441737867, −7.08242157238842790714464552373, −6.62330950254140969037261027991, −5.69373536355545884829545220101, −5.20091096263996294798150260812, −3.68542992275688587151208679266, −3.05175144147053173870901135048, −2.26351845946897083304766377945, −0.25278323720317297032762670499,
1.63163441214780115204665154170, 2.93951509813445098967979435560, 3.63106675412424160452059355394, 4.31622391799033002518636751681, 5.26704464773538140342161261171, 6.32983424169353980117017832560, 6.85590838206776665144981171837, 7.80991453847686654919145066784, 8.893339020884655077259278714152, 9.703796906551048901966168568015