L(s) = 1 | + (−36.6 − 36.6i)2-s + (−13.8 + 13.8i)3-s + 1.66e3i·4-s + (−721. + 3.04e3i)5-s + 1.01e3·6-s + (−1.38e4 − 1.38e4i)7-s + (2.33e4 − 2.33e4i)8-s + 5.86e4i·9-s + (1.37e5 − 8.49e4i)10-s − 2.57e5·11-s + (−2.29e4 − 2.29e4i)12-s + (1.07e5 − 1.07e5i)13-s + 1.01e6i·14-s + (−3.20e4 − 5.20e4i)15-s − 1.12e4·16-s + (−4.39e5 − 4.39e5i)17-s + ⋯ |
L(s) = 1 | + (−1.14 − 1.14i)2-s + (−0.0569 + 0.0569i)3-s + 1.62i·4-s + (−0.230 + 0.973i)5-s + 0.130·6-s + (−0.826 − 0.826i)7-s + (0.713 − 0.713i)8-s + 0.993i·9-s + (1.37 − 0.849i)10-s − 1.59·11-s + (−0.0923 − 0.0923i)12-s + (0.288 − 0.288i)13-s + 1.89i·14-s + (−0.0422 − 0.0685i)15-s − 0.0107·16-s + (−0.309 − 0.309i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.949i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(0.0725546 + 0.100553i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0725546 + 0.100553i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (721. - 3.04e3i)T \) |
good | 2 | \( 1 + (36.6 + 36.6i)T + 1.02e3iT^{2} \) |
| 3 | \( 1 + (13.8 - 13.8i)T - 5.90e4iT^{2} \) |
| 7 | \( 1 + (1.38e4 + 1.38e4i)T + 2.82e8iT^{2} \) |
| 11 | \( 1 + 2.57e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + (-1.07e5 + 1.07e5i)T - 1.37e11iT^{2} \) |
| 17 | \( 1 + (4.39e5 + 4.39e5i)T + 2.01e12iT^{2} \) |
| 19 | \( 1 + 1.46e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + (3.17e6 - 3.17e6i)T - 4.14e13iT^{2} \) |
| 29 | \( 1 - 1.87e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 1.47e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + (-1.77e7 - 1.77e7i)T + 4.80e15iT^{2} \) |
| 41 | \( 1 - 1.46e8T + 1.34e16T^{2} \) |
| 43 | \( 1 + (8.02e7 - 8.02e7i)T - 2.16e16iT^{2} \) |
| 47 | \( 1 + (1.20e8 + 1.20e8i)T + 5.25e16iT^{2} \) |
| 53 | \( 1 + (4.48e8 - 4.48e8i)T - 1.74e17iT^{2} \) |
| 59 | \( 1 - 2.00e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 4.13e7T + 7.13e17T^{2} \) |
| 67 | \( 1 + (-1.80e8 - 1.80e8i)T + 1.82e18iT^{2} \) |
| 71 | \( 1 + 3.07e8T + 3.25e18T^{2} \) |
| 73 | \( 1 + (1.88e9 - 1.88e9i)T - 4.29e18iT^{2} \) |
| 79 | \( 1 + 3.83e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + (-2.47e9 + 2.47e9i)T - 1.55e19iT^{2} \) |
| 89 | \( 1 - 3.22e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (2.24e8 + 2.24e8i)T + 7.37e19iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.76839894179693404927698170272, −20.12081285586125728249426684752, −19.06066998685258030529865929454, −17.90763000311153116120025609369, −16.05313019702372243812113042010, −13.28877606597857249822928924492, −10.98888133107929397842395333287, −10.15793776436543915289126541476, −7.69740493281476003312883576578, −2.82766553468071377367811853355,
0.12884671991971057629268711482, 6.00538086694833679324473316969, 8.238369916352422358927184627645, 9.605202129515795081405051467158, 12.61882099514438021896139125609, 15.42450014766179170428202850477, 16.22830535335301902730046224698, 17.85419465038530680595408698562, 19.03972873461781291980663339671, 20.91786949440419291049022172277