L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (0.935 − 0.540i)5-s − 0.999i·6-s + (1.55 − 2.14i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.540 − 0.935i)10-s + (−4.99 − 2.88i)11-s + (−0.499 − 0.866i)12-s + (0.235 + 3.59i)13-s + (0.275 − 2.63i)14-s − 1.08i·15-s + (−0.5 − 0.866i)16-s + (−1.54 + 2.68i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (0.418 − 0.241i)5-s − 0.408i·6-s + (0.587 − 0.809i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.170 − 0.295i)10-s + (−1.50 − 0.870i)11-s + (−0.144 − 0.249i)12-s + (0.0651 + 0.997i)13-s + (0.0735 − 0.703i)14-s − 0.278i·15-s + (−0.125 − 0.216i)16-s + (−0.375 + 0.650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56667 - 1.73582i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56667 - 1.73582i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.55 + 2.14i)T \) |
| 13 | \( 1 + (-0.235 - 3.59i)T \) |
good | 5 | \( 1 + (-0.935 + 0.540i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (4.99 + 2.88i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.54 - 2.68i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.93 + 3.42i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.46 - 4.27i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.05T + 29T^{2} \) |
| 31 | \( 1 + (3.33 + 1.92i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.06 - 1.19i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.58iT - 41T^{2} \) |
| 43 | \( 1 - 9.19T + 43T^{2} \) |
| 47 | \( 1 + (-0.297 + 0.171i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.09 - 8.82i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (9.94 + 5.73i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.01 - 3.49i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.8 - 6.27i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.4iT - 71T^{2} \) |
| 73 | \( 1 + (-6.41 - 3.70i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.98 - 3.44i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 + (-8.96 + 5.17i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.93iT - 97T^{2} \) |
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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
−10.92084534824692324817812556201, −9.764670019513119751860078130922, −8.811250776175027724561896348143, −7.73698446926511624712727670050, −7.00745720203885925699575331212, −5.73240821421144742825767049834, −4.97005138751842250261458940750, −3.70996227894735986395099268066, −2.50475907188947679748914969957, −1.19107364346317785511268681745,
2.34480543328714188314843478289, 3.09676635596626435943534369984, 4.82309137408653268731558087546, 5.18532493604721326272236287562, 6.25429875854973339914471550645, 7.65222879763261511487380641722, 8.107734653751344824675577278048, 9.311190592059085702397162604003, 10.23151900818252103923407773273, 10.90960410946971718080398675311