L(s) = 1 | + (−2.22 + 0.224i)5-s + (−1.41 + 1.41i)7-s + 3.46·11-s + (−0.317 + 0.317i)13-s + (−2.04 − 2.04i)17-s + 2.89·19-s + (0.449 − 0.449i)23-s + (4.89 − i)25-s + 3.55i·29-s − 4.09·31-s + (2.82 − 3.46i)35-s + (5.97 + 5.97i)37-s − 1.41i·41-s + (−2.89 + 2.89i)43-s + (−6.44 − 6.44i)47-s + ⋯ |
L(s) = 1 | + (−0.994 + 0.100i)5-s + (−0.534 + 0.534i)7-s + 1.04·11-s + (−0.0881 + 0.0881i)13-s + (−0.497 − 0.497i)17-s + 0.665·19-s + (0.0937 − 0.0937i)23-s + (0.979 − 0.200i)25-s + 0.659i·29-s − 0.736·31-s + (0.478 − 0.585i)35-s + (0.982 + 0.982i)37-s − 0.220i·41-s + (−0.442 + 0.442i)43-s + (−0.940 − 0.940i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5929788289\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5929788289\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.22 - 0.224i)T \) |
good | 7 | \( 1 + (1.41 - 1.41i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + (0.317 - 0.317i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.04 + 2.04i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.89T + 19T^{2} \) |
| 23 | \( 1 + (-0.449 + 0.449i)T - 23iT^{2} \) |
| 29 | \( 1 - 3.55iT - 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 + (-5.97 - 5.97i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (2.89 - 2.89i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.44 + 6.44i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.89 + 2.89i)T + 53iT^{2} \) |
| 59 | \( 1 - 6.29iT - 59T^{2} \) |
| 61 | \( 1 - 13.2iT - 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 + 12.8iT - 71T^{2} \) |
| 73 | \( 1 + (7.89 + 7.89i)T + 73iT^{2} \) |
| 79 | \( 1 - 14.1iT - 79T^{2} \) |
| 83 | \( 1 + (9.12 + 9.12i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 + (7.89 - 7.89i)T - 97iT^{2} \) |
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
−9.065061856102512325493681136868, −8.402429910937506819529500936094, −7.48376169355293856716727524054, −6.85231635758223842136364904455, −6.18281368325163367207243082830, −5.12511292834276025245506339237, −4.31246011232736168746722795639, −3.45656212231299830327546431694, −2.72805445060260205881641485041, −1.27548559096633997846033248493,
0.20907485590633986413081375866, 1.45310244924146603545918299405, 2.92685937277807527549619414040, 3.83703006840388329277178970484, 4.24756561456182621311916065209, 5.34153238307681653087254500966, 6.39203410855518120941493921496, 6.95555698619937226370898374517, 7.71844590252248598201018279449, 8.386519791534627224057312801904