L(s) = 1 | + (−1.09 − 0.899i)2-s + (−1.21 + 1.23i)3-s + (0.383 + 1.96i)4-s + (−0.726 − 2.70i)5-s + (2.43 − 0.254i)6-s + (−0.00424 − 0.00735i)7-s + (1.34 − 2.48i)8-s + (−0.0469 − 2.99i)9-s + (−1.64 + 3.61i)10-s + (0.804 − 3.00i)11-s + (−2.88 − 1.91i)12-s + (−1.72 − 6.42i)13-s + (−0.00197 + 0.0118i)14-s + (4.22 + 2.39i)15-s + (−3.70 + 1.50i)16-s + 3.37i·17-s + ⋯ |
L(s) = 1 | + (−0.771 − 0.635i)2-s + (−0.701 + 0.712i)3-s + (0.191 + 0.981i)4-s + (−0.324 − 1.21i)5-s + (0.994 − 0.104i)6-s + (−0.00160 − 0.00277i)7-s + (0.476 − 0.879i)8-s + (−0.0156 − 0.999i)9-s + (−0.519 + 1.14i)10-s + (0.242 − 0.904i)11-s + (−0.833 − 0.552i)12-s + (−0.477 − 1.78i)13-s + (−0.000528 + 0.00316i)14-s + (1.09 + 0.618i)15-s + (−0.926 + 0.376i)16-s + 0.819i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.281340 - 0.398152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.281340 - 0.398152i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.09 + 0.899i)T \) |
| 3 | \( 1 + (1.21 - 1.23i)T \) |
good | 5 | \( 1 + (0.726 + 2.70i)T + (-4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (0.00424 + 0.00735i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.804 + 3.00i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (1.72 + 6.42i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 3.37iT - 17T^{2} \) |
| 19 | \( 1 + (-1.35 - 1.35i)T + 19iT^{2} \) |
| 23 | \( 1 + (5.38 + 3.11i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.878 + 3.27i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-6.70 - 3.86i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.769 - 0.769i)T + 37iT^{2} \) |
| 41 | \( 1 + (2.64 - 4.58i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.99 + 1.60i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (0.0955 + 0.165i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.750 - 0.750i)T - 53iT^{2} \) |
| 59 | \( 1 + (9.08 - 2.43i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-9.85 - 2.64i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (7.77 - 2.08i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 1.02iT - 71T^{2} \) |
| 73 | \( 1 + 7.30iT - 73T^{2} \) |
| 79 | \( 1 + (-4.36 + 2.51i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.5 - 3.35i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + (-1.64 - 2.84i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.32660257376451632401086225431, −11.85577519653603966452015474998, −10.55078285053607828484268625246, −9.965323418192064658165584675374, −8.642039325865226009313184923904, −8.050301272052564070995174462120, −6.09013506437391052235670793866, −4.74302912485109059767867875025, −3.43235699726277995030531137475, −0.68907210192245372336824273432,
2.10137390828319798802481199637, 4.75085134633994741833770841456, 6.36537046092574656714349435042, 7.01019744920844915772258380391, 7.69005001003387697919900964240, 9.346716509541023265392256956439, 10.30851678117903258073895040638, 11.47440927005813299019027794251, 11.91373187400257293486436825010, 13.77613751805715985785699160589