L(s) = 1 | + (0.5 + 0.866i)3-s + 2·5-s + (−0.5 + 0.866i)7-s + (1 − 1.73i)9-s + (0.5 + 0.866i)11-s + (−1 + 3.46i)13-s + (1 + 1.73i)15-s + (−1.5 + 2.59i)17-s + (3.5 − 6.06i)19-s − 0.999·21-s + (0.5 + 0.866i)23-s − 25-s + 5·27-s + (−1.5 − 2.59i)29-s − 8·31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + 0.894·5-s + (−0.188 + 0.327i)7-s + (0.333 − 0.577i)9-s + (0.150 + 0.261i)11-s + (−0.277 + 0.960i)13-s + (0.258 + 0.447i)15-s + (−0.363 + 0.630i)17-s + (0.802 − 1.39i)19-s − 0.218·21-s + (0.104 + 0.180i)23-s − 0.200·25-s + 0.962·27-s + (−0.278 − 0.482i)29-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41858 + 0.370376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41858 + 0.370376i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.5 + 9.52i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.5 - 4.33i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.51948253416546243222657414981, −11.49043630666685630724640364067, −10.31467387893329122501653361027, −9.276180823173705844397508591984, −9.083276185073369297561054719299, −7.25144494024617235415210610250, −6.27795133156311564560214669821, −5.00741929795336241961990309039, −3.69501604350281363573313018201, −2.07162678969490421927973870335,
1.67956063046324334839416555804, 3.20070042369306448075093964637, 5.01190754974758562433015422114, 6.09785890585220915288897168901, 7.33052643195601477090663844057, 8.144827716681293545193976597937, 9.529813331501592213481366593951, 10.20327905523708994639613019851, 11.29251388592670308488683785788, 12.65054342160159726881332198973