L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (3.72 − 2.15i)5-s − 0.999·6-s + (−1.43 − 2.22i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (2.15 − 3.72i)10-s + (−3.27 + 0.553i)11-s + (−0.866 + 0.499i)12-s + 1.00·13-s + (−2.35 − 1.21i)14-s − 4.30·15-s + (−0.5 − 0.866i)16-s + (−1.66 + 2.88i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (1.66 − 0.961i)5-s − 0.408·6-s + (−0.541 − 0.840i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.680 − 1.17i)10-s + (−0.985 + 0.166i)11-s + (−0.249 + 0.144i)12-s + 0.277·13-s + (−0.628 − 0.323i)14-s − 1.11·15-s + (−0.125 − 0.216i)16-s + (−0.404 + 0.700i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26857 - 1.52186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26857 - 1.52186i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (1.43 + 2.22i)T \) |
| 11 | \( 1 + (3.27 - 0.553i)T \) |
good | 5 | \( 1 + (-3.72 + 2.15i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 + (1.66 - 2.88i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.61 - 2.79i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.86 - 4.96i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.74iT - 29T^{2} \) |
| 31 | \( 1 + (2.77 + 1.60i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.26 - 2.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 - 4.14iT - 43T^{2} \) |
| 47 | \( 1 + (-2.77 + 1.60i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.65 + 9.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.98 - 1.72i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.42 - 12.8i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.165 + 0.286i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.84T + 71T^{2} \) |
| 73 | \( 1 + (7.44 - 12.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.9 + 8.04i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 + (2.41 - 1.39i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.3iT - 97T^{2} \) |
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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
−10.66274680303681305396086519201, −10.06128066895805694261442902231, −9.379735961882063028577927754160, −8.022255120858065938167958262409, −6.74008133590502370217159586195, −5.82198671967374660498498760293, −5.27147478881339649717469536019, −4.07110040036181348121289915946, −2.37227829625718092234294480479, −1.15470396450481984446168800753,
2.38301425617002228889269251651, 3.12315802861782027370430773479, 5.08216990957357263411587484865, 5.57609057297829348882989956110, 6.49611376831689355533580934715, 7.10679707626689765449737713997, 8.845392110549615699224830832640, 9.536530358205605888663484331008, 10.62249738777075767493545391095, 11.03157060517410472022155147547