L(s) = 1 | + (1.41 − 0.0681i)2-s + (1.99 − 0.192i)4-s + (−2.08 + 3.61i)5-s + (−2.39 + 1.12i)7-s + (2.79 − 0.407i)8-s + (−2.70 + 5.25i)10-s + (0.855 + 1.48i)11-s − 1.54·13-s + (−3.30 + 1.74i)14-s + (3.92 − 0.766i)16-s + (−2.02 + 1.16i)17-s + (6.09 + 3.52i)19-s + (−3.45 + 7.60i)20-s + (1.30 + 2.03i)22-s + (0.406 + 0.234i)23-s + ⋯ |
L(s) = 1 | + (0.998 − 0.0482i)2-s + (0.995 − 0.0963i)4-s + (−0.933 + 1.61i)5-s + (−0.905 + 0.423i)7-s + (0.989 − 0.144i)8-s + (−0.854 + 1.66i)10-s + (0.257 + 0.446i)11-s − 0.427·13-s + (−0.884 + 0.466i)14-s + (0.981 − 0.191i)16-s + (−0.490 + 0.282i)17-s + (1.39 + 0.807i)19-s + (−0.773 + 1.69i)20-s + (0.279 + 0.433i)22-s + (0.0846 + 0.0488i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0684 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0684 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.47386 + 1.37622i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47386 + 1.37622i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 + 0.0681i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.39 - 1.12i)T \) |
good | 5 | \( 1 + (2.08 - 3.61i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.855 - 1.48i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 + (2.02 - 1.16i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.09 - 3.52i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.406 - 0.234i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.33iT - 29T^{2} \) |
| 31 | \( 1 + (-1.58 - 2.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.74 - 4.47i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.31iT - 41T^{2} \) |
| 43 | \( 1 + 3.42T + 43T^{2} \) |
| 47 | \( 1 + (-2.95 + 5.11i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.35 + 0.781i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.26 + 3.04i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.55 + 7.89i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.73 - 6.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.49iT - 71T^{2} \) |
| 73 | \( 1 + (12.5 - 7.26i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.46 + 0.843i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.72iT - 83T^{2} \) |
| 89 | \( 1 + (1.83 + 1.06i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.95iT - 97T^{2} \) |
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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
−11.49248971556877364937360444200, −10.33753772902810566936166837841, −9.798462681811529201282720417114, −8.066713373198754310103498501332, −7.11891091166417073030009935138, −6.63863994496862265140747503262, −5.61572382594772345287262871060, −4.12371512019760064398512998241, −3.31637250547902977439967717247, −2.46291215114026826134587511171,
0.902935437191133450352652682727, 3.00809096736607721311945041579, 4.08918773814015951648718936908, 4.82571618683517201052645172953, 5.80670018514376064936002364989, 7.06430084019912132731270011933, 7.77864353069223458757979729366, 8.928324277685403016458935017425, 9.731578720210418941881268747111, 11.14572343634208252532220921716