L(s) = 1 | − 2-s + (−0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (1.5 + 2.59i)11-s + (−0.5 + 0.866i)12-s + (−1 − 1.73i)13-s + (−0.5 + 0.866i)14-s − 0.999·15-s + 16-s + (−3 + 5.19i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (0.188 − 0.327i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.452 + 0.783i)11-s + (−0.144 + 0.249i)12-s + (−0.277 − 0.480i)13-s + (−0.133 + 0.231i)14-s − 0.258·15-s + 0.250·16-s + (−0.727 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.542732 + 0.719779i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.542732 + 0.719779i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (3.5 + 4.33i)T \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.5 - 12.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6 - 10.3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 19T + 97T^{2} \) |
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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
−10.30820555418554570971033039596, −9.640044815890584304711187261602, −8.717754513599062075332712427137, −7.910285913194898528289174964703, −6.84043576405043844082164386713, −6.28211456499897401376617156653, −5.03606763210404721564561377676, −4.06668902048669258430836161170, −2.81436069562022185395438184424, −1.43845497735194352975432962682,
0.58192753142310427380954381581, 1.91641237494418434948207602502, 3.06703750625511754457083983035, 4.72185246044900715789996531479, 5.53288856421949049965336521547, 6.74909678893546006042465072379, 7.06196729034940255194392215070, 8.470043355380162917765803633227, 8.821014470046287667059007273344, 9.636117515485580174543391905415