L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.292 − 0.507i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.585·6-s − 0.999·8-s + (1.32 − 2.30i)9-s + (0.499 + 0.866i)10-s + (−2.41 − 4.18i)11-s + (−0.292 + 0.507i)12-s + 0.828·13-s + 0.585·15-s + (−0.5 + 0.866i)16-s + (−2.70 − 4.68i)17-s + (−1.32 − 2.30i)18-s + (−1.70 + 2.95i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.169 − 0.292i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 0.239·6-s − 0.353·8-s + (0.442 − 0.766i)9-s + (0.158 + 0.273i)10-s + (−0.727 − 1.26i)11-s + (−0.0845 + 0.146i)12-s + 0.229·13-s + 0.151·15-s + (−0.125 + 0.216i)16-s + (−0.656 − 1.13i)17-s + (−0.313 − 0.542i)18-s + (−0.391 + 0.678i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.440246 - 1.16435i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.440246 - 1.16435i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.292 + 0.507i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (2.41 + 4.18i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 + (2.70 + 4.68i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.70 - 2.95i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.41 + 5.91i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + (1.41 + 2.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.82 - 3.16i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 + (5.41 - 9.37i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.24 - 9.08i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.70 + 9.88i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.65 + 11.5i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.82 - 8.36i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + (-3.29 - 5.70i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.585 + 1.01i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + (-6.36 + 11.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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
−10.97956873325605296703037616845, −9.888333504994092966995338639249, −8.940283817899564388455743234917, −7.936771867383309706891939166051, −6.72736538157589128210734524460, −5.99150274161858809216318371203, −4.74212017091794765072573619744, −3.56968288990767405742367342808, −2.55761865698884079924560305099, −0.68731878865112363732629202782,
2.08890214597006709681330387013, 3.86795775579693543455859974492, 4.76663202380792385127221259651, 5.45401877792578967598540482494, 6.82453946204241736379610225322, 7.57108997768208190963377909849, 8.468347356648939815510814060876, 9.465009541768465555584453351726, 10.45408182656366237474136606326, 11.22041229535771309585642947499