L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.929 − 0.675i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (−0.355 + 1.09i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.519 − 1.59i)9-s − 0.999·10-s + (3.22 − 0.780i)11-s + 1.14·12-s + (1.17 + 3.61i)13-s + (−0.809 − 0.587i)14-s + (−0.929 + 0.675i)15-s + (0.309 − 0.951i)16-s + (2.00 − 6.16i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.536 − 0.389i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (−0.144 + 0.446i)6-s + (0.305 − 0.222i)7-s + (0.286 + 0.207i)8-s + (−0.173 − 0.532i)9-s − 0.316·10-s + (0.971 − 0.235i)11-s + 0.331·12-s + (0.325 + 1.00i)13-s + (−0.216 − 0.157i)14-s + (−0.240 + 0.174i)15-s + (0.0772 − 0.237i)16-s + (0.485 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.259488 - 1.00365i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.259488 - 1.00365i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (-3.22 + 0.780i)T \) |
good | 3 | \( 1 + (0.929 + 0.675i)T + (0.927 + 2.85i)T^{2} \) |
| 13 | \( 1 + (-1.17 - 3.61i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.00 + 6.16i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.16 + 1.57i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 2.65T + 23T^{2} \) |
| 29 | \( 1 + (0.313 - 0.227i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.85 + 5.72i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.93 - 1.40i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (7.66 + 5.57i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.66T + 43T^{2} \) |
| 47 | \( 1 + (-3.62 - 2.63i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.57 + 4.84i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.83 - 2.78i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.60 + 11.0i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 5.71T + 67T^{2} \) |
| 71 | \( 1 + (-1.02 + 3.16i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (12.1 - 8.82i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.86 + 5.74i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.44 - 10.6i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 5.87T + 89T^{2} \) |
| 97 | \( 1 + (1.02 + 3.14i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.856274759672038027039409084524, −9.170843470656115353544877056221, −8.574793400910922314519733376648, −7.23463440646910579950834670324, −6.55575625592699210610243668337, −5.41527078363167880647961864916, −4.40266959609217051734901214528, −3.35882045884041662450073943445, −1.77954600747846789035589353804, −0.65052899070501946916026009885,
1.62150321867411606631425156849, 3.40556581968031769065451451346, 4.54920242598209250854149289369, 5.57133720808075163093377264814, 6.14616740412755423089659072184, 7.15572454411531309058152591396, 8.203066021251277770394225232574, 8.746623395791444314674062260616, 10.05274164971746665404574720427, 10.47613378870921301943990773847