L(s) = 1 | + 2-s + (0.482 − 1.66i)3-s + 4-s − 0.885i·5-s + (0.482 − 1.66i)6-s − i·7-s + 8-s + (−2.53 − 1.60i)9-s − 0.885i·10-s + (−1.14 − 3.11i)11-s + (0.482 − 1.66i)12-s + 1.32i·13-s − i·14-s + (−1.47 − 0.426i)15-s + 16-s + 1.35·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.278 − 0.960i)3-s + 0.5·4-s − 0.395i·5-s + (0.196 − 0.679i)6-s − 0.377i·7-s + 0.353·8-s + (−0.845 − 0.534i)9-s − 0.279i·10-s + (−0.345 − 0.938i)11-s + (0.139 − 0.480i)12-s + 0.368i·13-s − 0.267i·14-s + (−0.380 − 0.110i)15-s + 0.250·16-s + 0.329·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0705 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0705 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62373 - 1.51290i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62373 - 1.51290i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.482 + 1.66i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (1.14 + 3.11i)T \) |
good | 5 | \( 1 + 0.885iT - 5T^{2} \) |
| 13 | \( 1 - 1.32iT - 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 - 4.67iT - 19T^{2} \) |
| 23 | \( 1 + 1.08iT - 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 - 0.592T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 - 0.991iT - 43T^{2} \) |
| 47 | \( 1 + 0.204iT - 47T^{2} \) |
| 53 | \( 1 + 1.07iT - 53T^{2} \) |
| 59 | \( 1 + 5.25iT - 59T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 14.0iT - 71T^{2} \) |
| 73 | \( 1 - 6.28iT - 73T^{2} \) |
| 79 | \( 1 + 5.69iT - 79T^{2} \) |
| 83 | \( 1 + 9.95T + 83T^{2} \) |
| 89 | \( 1 - 6.72iT - 89T^{2} \) |
| 97 | \( 1 + 3.90T + 97T^{2} \) |
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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
−11.10327418223184207634399973307, −10.07707265441890784061231750972, −8.702222050846856895410847189069, −8.073075790196821816182052194786, −7.02789113688931291222164695872, −6.16214172904011766634168665304, −5.21607166352621020903235382883, −3.82815381911416350046807047522, −2.71679137826691718320936204508, −1.17959187559396655041864071780,
2.43453743897452011011300686215, 3.31636658711371993059426576388, 4.62294650114551487269675444137, 5.24299816470281865079068700441, 6.48365970905091706268982572933, 7.55156784540708680365551727203, 8.634406931219916645685494022532, 9.650383277293898305798120035472, 10.45103003049974821926285192707, 11.16238556183187092243283244686