| L(s) = 1 | + (−1.40 − 0.130i)2-s + (0.342 − 0.939i)3-s + (1.96 + 0.367i)4-s + (−0.0410 + 0.00723i)5-s + (−0.604 + 1.27i)6-s + (−1.18 + 2.05i)7-s + (−2.72 − 0.773i)8-s + (−0.766 − 0.642i)9-s + (0.0587 − 0.00483i)10-s + (4.23 − 2.44i)11-s + (1.01 − 1.72i)12-s + (1.16 + 3.20i)13-s + (1.93 − 2.73i)14-s + (−0.00723 + 0.0410i)15-s + (3.73 + 1.44i)16-s + (2.30 − 1.93i)17-s + ⋯ |
| L(s) = 1 | + (−0.995 − 0.0922i)2-s + (0.197 − 0.542i)3-s + (0.982 + 0.183i)4-s + (−0.0183 + 0.00323i)5-s + (−0.246 + 0.522i)6-s + (−0.448 + 0.776i)7-s + (−0.961 − 0.273i)8-s + (−0.255 − 0.214i)9-s + (0.0185 − 0.00152i)10-s + (1.27 − 0.738i)11-s + (0.293 − 0.497i)12-s + (0.323 + 0.889i)13-s + (0.518 − 0.732i)14-s + (−0.00186 + 0.0105i)15-s + (0.932 + 0.361i)16-s + (0.558 − 0.468i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.995708 - 0.191676i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.995708 - 0.191676i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.130i)T \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 19 | \( 1 + (-4.20 - 1.13i)T \) |
| good | 5 | \( 1 + (0.0410 - 0.00723i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.18 - 2.05i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.23 + 2.44i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.16 - 3.20i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.30 + 1.93i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.281 - 1.59i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.88 + 4.62i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.72 + 8.18i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.19iT - 37T^{2} \) |
| 41 | \( 1 + (0.487 + 0.177i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-10.2 + 1.80i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.77 - 4.01i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (2.10 + 0.371i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (8.37 + 9.98i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.262 - 0.0462i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.74 + 10.4i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.81 - 10.2i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (2.97 + 1.08i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (10.9 + 3.99i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.53 - 4.34i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (14.5 - 5.28i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (4.09 - 3.43i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.23606449981927980882027312093, −9.561722989577390426119609436140, −9.441716077588730359237879724000, −8.359551297083765294487849973113, −7.53459718173965466853420586576, −6.39391208434062512710800620280, −5.89763600934763543030626105651, −3.75648104849111754845251770816, −2.57878377798533278659276766723, −1.16166898548054783163141073780,
1.17234184484031785601622596747, 3.01588479343033926840568542751, 4.09482795814619554628631538893, 5.64204972215129302169576930382, 6.74302469096780528735428576905, 7.53216566427557812906674604462, 8.567456144035239679633612163876, 9.410501754766690245762173073175, 10.18430842351401000585325301785, 10.67788745711874628361827465821
