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
−10.47613378870921301943990773847, −10.05274164971746665404574720427, −8.746623395791444314674062260616, −8.203066021251277770394225232574, −7.15572454411531309058152591396, −6.14616740412755423089659072184, −5.57133720808075163093377264814, −4.54920242598209250854149289369, −3.40556581968031769065451451346, −1.62150321867411606631425156849,
0.65052899070501946916026009885, 1.77954600747846789035589353804, 3.35882045884041662450073943445, 4.40266959609217051734901214528, 5.41527078363167880647961864916, 6.55575625592699210610243668337, 7.23463440646910579950834670324, 8.574793400910922314519733376648, 9.170843470656115353544877056221, 9.856274759672038027039409084524