L(s) = 1 | + (−1.32 − 2.29i)3-s + (1.5 + 0.866i)5-s + 2.64·7-s + (−2 + 3.46i)9-s + (3.96 − 2.29i)11-s + 3.46i·13-s − 4.58i·15-s + (4.5 − 2.59i)17-s + (−1.32 + 2.29i)19-s + (−3.50 − 6.06i)21-s + (−3.96 − 2.29i)23-s + (−1 − 1.73i)25-s + 2.64·27-s + (−1.32 − 2.29i)31-s + (−10.5 − 6.06i)33-s + ⋯ |
L(s) = 1 | + (−0.763 − 1.32i)3-s + (0.670 + 0.387i)5-s + 0.999·7-s + (−0.666 + 1.15i)9-s + (1.19 − 0.690i)11-s + 0.960i·13-s − 1.18i·15-s + (1.09 − 0.630i)17-s + (−0.303 + 0.525i)19-s + (−0.763 − 1.32i)21-s + (−0.827 − 0.477i)23-s + (−0.200 − 0.346i)25-s + 0.509·27-s + (−0.237 − 0.411i)31-s + (−1.82 − 1.05i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17341 - 0.780537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17341 - 0.780537i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
good | 3 | \( 1 + (1.32 + 2.29i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.96 + 2.29i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (-4.5 + 2.59i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.32 - 2.29i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.96 + 2.29i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (1.32 + 2.29i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 9.16iT - 43T^{2} \) |
| 47 | \( 1 + (3.96 - 6.87i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.96 - 6.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 + 0.866i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.96 - 2.29i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.16iT - 71T^{2} \) |
| 73 | \( 1 + (4.5 - 2.59i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.96 + 2.29i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-1.5 - 0.866i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.46iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.30202691144139223473357634401, −10.18490235655004829236480336806, −9.038066239943585338842750324741, −7.968956888757223828848834682783, −7.10557095564381856770075766537, −6.19513407035626156248461706138, −5.65290601887123994081081344487, −4.13151654179932512764445327417, −2.19938048181232045180498501624, −1.19941812025610903381501645613,
1.55339075057864645371120605138, 3.63663572475795314782868834463, 4.66167651489746545001916458359, 5.37756444702865522295950839197, 6.20546052560161948639956514912, 7.71247425798506135900843527402, 8.800816327665565868240854950826, 9.778985154469176778963909054921, 10.19156367298014280579871131753, 11.22367469218059324456310604118