L(s) = 1 | + (−0.866 − 1.5i)3-s + (1.73 − 2i)7-s + (−1.5 + 2.59i)9-s + (−4.5 − 2.59i)11-s + (−2.59 − 1.5i)17-s + (−1.5 + 0.866i)19-s + (−4.5 − 0.866i)21-s + (2.59 + 4.5i)23-s + 5.19·27-s + (−1.5 − 0.866i)31-s + 9i·33-s + (−6.06 + 3.5i)37-s − 6·41-s + 4i·43-s + (2.59 − 1.5i)47-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.866i)3-s + (0.654 − 0.755i)7-s + (−0.5 + 0.866i)9-s + (−1.35 − 0.783i)11-s + (−0.630 − 0.363i)17-s + (−0.344 + 0.198i)19-s + (−0.981 − 0.188i)21-s + (0.541 + 0.938i)23-s + 1.00·27-s + (−0.269 − 0.155i)31-s + 1.56i·33-s + (−0.996 + 0.575i)37-s − 0.937·41-s + 0.609i·43-s + (0.378 − 0.218i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 11 | \( 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.59 - 4.5i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.06 - 3.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (-2.59 + 1.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.59 - 4.5i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (10.5 - 6.06i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-6.06 + 10.5i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−8.255850473822746027398165468527, −7.79964466275003963683839594660, −7.08768857594406002828074417605, −6.27358226085314040819139179374, −5.29924512609774097012246826055, −4.82852255630888881980074590133, −3.47028954183801285644802214721, −2.37983018282427193253100459633, −1.27165119230246777618525923499, 0,
1.99052856617929217683778745171, 2.91387404381354825588418963718, 4.17011460602346200596067350531, 4.96469154143409362372008525671, 5.38044841727998486059698305274, 6.38557558593624985907956757284, 7.26867164338789418191535056639, 8.355644795566042981984506649681, 8.800199552324713498011604610079