L(s) = 1 | + (1.62 − 2.81i)3-s + (0.235 − 0.407i)5-s − 3.30·7-s + (−3.77 − 6.54i)9-s − 1.47·11-s + (−1.41 − 2.45i)13-s + (−0.764 − 1.32i)15-s + (−3.35 + 5.81i)17-s + (0.206 − 4.35i)19-s + (−5.37 + 9.30i)21-s + (−0.235 − 0.407i)23-s + (2.38 + 4.13i)25-s − 14.8·27-s + (2.48 + 4.30i)29-s + 6.74·31-s + ⋯ |
L(s) = 1 | + (0.937 − 1.62i)3-s + (0.105 − 0.182i)5-s − 1.25·7-s + (−1.25 − 2.18i)9-s − 0.443·11-s + (−0.393 − 0.681i)13-s + (−0.197 − 0.341i)15-s + (−0.814 + 1.41i)17-s + (0.0472 − 0.998i)19-s + (−1.17 + 2.03i)21-s + (−0.0490 − 0.0849i)23-s + (0.477 + 0.827i)25-s − 2.84·27-s + (0.461 + 0.799i)29-s + 1.21·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.106893521\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.106893521\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-0.206 + 4.35i)T \) |
good | 3 | \( 1 + (-1.62 + 2.81i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.235 + 0.407i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + (1.41 + 2.45i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.35 - 5.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.235 + 0.407i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.48 - 4.30i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 + (-0.250 + 0.434i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.94 + 3.37i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.95 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.35 + 4.08i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.62 + 6.27i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.26 + 10.8i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.316 + 0.548i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.88 + 3.27i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.97 + 3.41i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.41 + 7.65i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.14T + 83T^{2} \) |
| 89 | \( 1 + (1.47 + 2.55i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0293 - 0.0507i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.975851118617173666788278938713, −8.463040824806342509041252171738, −7.63923961591255830269658555448, −6.66702712141911044040501264093, −6.46339705750386716707840284480, −5.17657214166989618397978478459, −3.51183738828040130706406227892, −2.85132993749347758760712683658, −1.83992054525733159650414014740, −0.38437937399979414730275881956,
2.59662970464139153501344751022, 2.97339254735079609012381496237, 4.17740139947356580544581247697, 4.72681396147408135737281320050, 5.90436160592377187847775757370, 6.89794142546467355075971742395, 7.997948063729415791751874381219, 8.807666685716945763599630177442, 9.623802464000007022092304154669, 9.865577952020762489429156813231