L(s) = 1 | + (0.515 − 0.892i)5-s + (1.55 − 2.14i)7-s + (−0.792 − 1.37i)11-s + 5.04·13-s + (2.58 + 4.47i)17-s + (−0.392 + 0.680i)19-s + (−2.93 + 5.07i)23-s + (1.96 + 3.40i)25-s + 8.89·29-s + (0.575 + 0.996i)31-s + (−1.10 − 2.49i)35-s + (4.07 − 7.06i)37-s − 7.74·41-s − 2.53·43-s + (4.24 − 7.35i)47-s + ⋯ |
L(s) = 1 | + (0.230 − 0.399i)5-s + (0.588 − 0.808i)7-s + (−0.239 − 0.414i)11-s + 1.40·13-s + (0.626 + 1.08i)17-s + (−0.0901 + 0.156i)19-s + (−0.611 + 1.05i)23-s + (0.393 + 0.681i)25-s + 1.65·29-s + (0.103 + 0.179i)31-s + (−0.187 − 0.421i)35-s + (0.670 − 1.16i)37-s − 1.20·41-s − 0.386·43-s + (0.619 − 1.07i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.201906425\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.201906425\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.55 + 2.14i)T \) |
good | 5 | \( 1 + (-0.515 + 0.892i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.792 + 1.37i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 + (-2.58 - 4.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.392 - 0.680i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.93 - 5.07i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.89T + 29T^{2} \) |
| 31 | \( 1 + (-0.575 - 0.996i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.07 + 7.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7.74T + 41T^{2} \) |
| 43 | \( 1 + 2.53T + 43T^{2} \) |
| 47 | \( 1 + (-4.24 + 7.35i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.41 + 4.17i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.93 - 3.35i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.82 + 8.35i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.837 - 1.45i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + (-3.04 - 5.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.15 + 7.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + (6.69 - 11.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−8.670483383023096172463384464660, −8.329683166252026508876717506646, −7.54372663728064898948359059329, −6.55133948552479957219784378293, −5.77877928911327608770450024795, −5.03891966644152950069233382226, −3.94940872543825876171662985051, −3.40113008062538139626128033886, −1.78238360323112623639206820727, −0.980027760943562536926010481942,
1.11564810925173207104951502788, 2.40526564907108299572570650382, 3.07680635512588918684338007663, 4.42086482656751609872853354313, 5.05597508528773668418643829364, 6.13688902037662803828947624806, 6.53550072486982352839339765284, 7.68274298860313635193884333845, 8.399950601008314682388452319534, 8.909788962153003972726898695788