L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.24 − 1.20i)3-s + (−0.499 + 0.866i)4-s + (0.0916 − 0.0529i)5-s + (−1.66 − 0.477i)6-s + (−2.29 − 1.31i)7-s + 0.999·8-s + (0.105 − 2.99i)9-s + (−0.0916 − 0.0529i)10-s + (0.603 − 1.04i)11-s + (0.418 + 1.68i)12-s + (−3.60 − 0.175i)13-s + (0.0147 + 2.64i)14-s + (0.0505 − 0.176i)15-s + (−0.5 − 0.866i)16-s + (1.14 − 1.97i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.719 − 0.694i)3-s + (−0.249 + 0.433i)4-s + (0.0409 − 0.0236i)5-s + (−0.679 − 0.195i)6-s + (−0.868 − 0.495i)7-s + 0.353·8-s + (0.0353 − 0.999i)9-s + (−0.0289 − 0.0167i)10-s + (0.182 − 0.315i)11-s + (0.120 + 0.485i)12-s + (−0.998 − 0.0485i)13-s + (0.00394 + 0.707i)14-s + (0.0130 − 0.0454i)15-s + (−0.125 − 0.216i)16-s + (0.276 − 0.479i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.175056 - 1.05813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.175056 - 1.05813i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.24 + 1.20i)T \) |
| 7 | \( 1 + (2.29 + 1.31i)T \) |
| 13 | \( 1 + (3.60 + 0.175i)T \) |
good | 5 | \( 1 + (-0.0916 + 0.0529i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.603 + 1.04i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.14 + 1.97i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.82 + 3.15i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.845 + 0.488i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.03iT - 29T^{2} \) |
| 31 | \( 1 + (-0.610 + 1.05i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.01 + 1.16i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.417iT - 41T^{2} \) |
| 43 | \( 1 - 4.90T + 43T^{2} \) |
| 47 | \( 1 + (-1.47 + 0.854i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.77 + 1.02i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.04 - 5.21i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.67 - 5.00i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.65 - 3.26i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.72T + 71T^{2} \) |
| 73 | \( 1 + (3.72 - 6.44i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.04 + 8.74i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 + (-14.7 + 8.48i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−10.17632855796760017204015107307, −9.513012644642818913566040333181, −8.835832248744330662685596125521, −7.65807261683875885365463742493, −7.11853869993067684718601301330, −5.98702532710896065641720017870, −4.33311734993307302969491986559, −3.22111246071999708449834009120, −2.31176514919982493322469723389, −0.61978000348222096193365407396,
2.20403102979667474181754653756, 3.49166612243848754399199228057, 4.65243521732860809227050919615, 5.70506569229844442387089905381, 6.75768374650441164502374567587, 7.77059120082135008721798988891, 8.607559853043472119408203665498, 9.465135758649933656709268845191, 9.972620791685662462513645488354, 10.77541431095373230560768970030