| L(s) = 1 | + (0.272 − 1.38i)2-s + (−0.881 + 0.471i)3-s + (−1.85 − 0.757i)4-s + (1.21 + 1.48i)5-s + (0.413 + 1.35i)6-s + (1.92 − 1.28i)7-s + (−1.55 + 2.36i)8-s + (0.555 − 0.831i)9-s + (2.39 − 1.28i)10-s + (0.740 − 0.224i)11-s + (1.98 − 0.204i)12-s + (1.85 + 1.52i)13-s + (−1.26 − 3.02i)14-s + (−1.77 − 0.734i)15-s + (2.85 + 2.80i)16-s + (5.13 − 2.12i)17-s + ⋯ |
| L(s) = 1 | + (0.192 − 0.981i)2-s + (−0.509 + 0.272i)3-s + (−0.925 − 0.378i)4-s + (0.544 + 0.663i)5-s + (0.168 + 0.552i)6-s + (0.728 − 0.486i)7-s + (−0.550 + 0.835i)8-s + (0.185 − 0.277i)9-s + (0.756 − 0.406i)10-s + (0.223 − 0.0677i)11-s + (0.574 − 0.0590i)12-s + (0.514 + 0.422i)13-s + (−0.336 − 0.808i)14-s + (−0.458 − 0.189i)15-s + (0.713 + 0.700i)16-s + (1.24 − 0.516i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22773 - 0.683227i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22773 - 0.683227i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.272 + 1.38i)T \) |
| 3 | \( 1 + (0.881 - 0.471i)T \) |
| good | 5 | \( 1 + (-1.21 - 1.48i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.92 + 1.28i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.740 + 0.224i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-1.85 - 1.52i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-5.13 + 2.12i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.262 + 2.66i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-3.99 + 0.795i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (0.385 - 1.27i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (4.74 + 4.74i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.647 + 0.0637i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-1.92 - 9.68i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (3.77 + 2.01i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (1.55 + 3.75i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-3.62 - 11.9i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (0.544 - 0.447i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-2.90 - 5.43i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-3.64 - 6.82i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-2.20 - 3.30i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (10.6 + 7.13i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.19 + 5.30i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (6.92 - 0.682i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-4.65 - 0.926i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (12.0 + 12.0i)T + 97iT^{2} \) |
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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
−11.20080054813384850908730306744, −10.46940134906880837917390325317, −9.714161434988379539357208875055, −8.756072739307303592155155644958, −7.39744860287050563174161329635, −6.15499827359888754310689732478, −5.14310967218406767310743373085, −4.12472339223189437008057234465, −2.86089333820631305343592888240, −1.27209025541154814075135612021,
1.39955655758618270914173523560, 3.67912988150622890685612375000, 5.28377870368488974054659857371, 5.41773832737115443388274805194, 6.61277245315707721707836741028, 7.77230389815220012156642565105, 8.521013525545924238992723123997, 9.406846851194457128543871954494, 10.49638688291169631110994869105, 11.72557929504302191998847817290
