L(s) = 1 | + (−3.93 − 2.27i)2-s + (−2.24 + 4.68i)3-s + (6.33 + 10.9i)4-s + (−5.80 + 10.0i)5-s + (19.4 − 13.3i)6-s + (−18.4 − 2.09i)7-s − 21.1i·8-s + (−16.9 − 21.0i)9-s + (45.6 − 26.3i)10-s + (15.5 − 8.95i)11-s + (−65.5 + 5.10i)12-s + 62.4i·13-s + (67.6 + 50.0i)14-s + (−34.1 − 49.7i)15-s + (2.48 − 4.30i)16-s + (10.7 + 18.5i)17-s + ⋯ |
L(s) = 1 | + (−1.39 − 0.803i)2-s + (−0.431 + 0.902i)3-s + (0.791 + 1.37i)4-s + (−0.518 + 0.898i)5-s + (1.32 − 0.909i)6-s + (−0.993 − 0.112i)7-s − 0.936i·8-s + (−0.627 − 0.778i)9-s + (1.44 − 0.833i)10-s + (0.425 − 0.245i)11-s + (−1.57 + 0.122i)12-s + 1.33i·13-s + (1.29 + 0.955i)14-s + (−0.587 − 0.855i)15-s + (0.0388 − 0.0673i)16-s + (0.152 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.160556 + 0.232316i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.160556 + 0.232316i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.24 - 4.68i)T \) |
| 7 | \( 1 + (18.4 + 2.09i)T \) |
good | 2 | \( 1 + (3.93 + 2.27i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (5.80 - 10.0i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-15.5 + 8.95i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 62.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-10.7 - 18.5i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-9.50 - 5.48i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (59.8 + 34.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 265. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-8.85 + 5.11i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (20.8 - 36.0i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 31.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 224.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (81.8 - 141. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-456. + 263. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (205. + 356. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-223. - 129. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (161. + 280. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 45.4iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (486. - 281. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (144. - 250. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 448.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (280. - 486. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 214. iT - 9.12e5T^{2} \) |
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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
−18.30592506851247702144117125683, −16.84747267888951757217065231592, −16.12882113672643134015040914487, −14.51201753215478167686753153561, −12.01993489823792389850137882039, −11.03409899991445826579799865348, −10.01358006164936783452050139361, −8.909722926072997755232074585116, −6.78596493212918899473536583037, −3.45078836605789536516177156316,
0.49993227456804802842988061035, 5.95681458648031482662014012012, 7.44370686395010856201929217494, 8.557991898623335540987290668631, 10.05744008269523393019179300667, 12.00611241831087813444493333860, 13.18395901570118084144886582583, 15.42610772620412333139074999369, 16.41436636276251413368248785633, 17.28534975875386268454386139840