L(s) = 1 | + (0.866 − 0.5i)2-s + (1.11 + 0.642i)3-s + (0.499 − 0.866i)4-s + (0.642 − 0.766i)5-s + 1.28·6-s + (0.642 + 0.766i)7-s − 0.999i·8-s + (0.326 + 0.565i)9-s + (0.173 − 0.984i)10-s + (1.11 − 0.642i)12-s + i·13-s + (0.939 + 0.342i)14-s + (1.20 − 0.439i)15-s + (−0.5 − 0.866i)16-s + (−1.70 − 0.984i)17-s + (0.565 + 0.326i)18-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (1.11 + 0.642i)3-s + (0.499 − 0.866i)4-s + (0.642 − 0.766i)5-s + 1.28·6-s + (0.642 + 0.766i)7-s − 0.999i·8-s + (0.326 + 0.565i)9-s + (0.173 − 0.984i)10-s + (1.11 − 0.642i)12-s + i·13-s + (0.939 + 0.342i)14-s + (1.20 − 0.439i)15-s + (−0.5 − 0.866i)16-s + (−1.70 − 0.984i)17-s + (0.565 + 0.326i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.443737167\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.443737167\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.642 + 0.766i)T \) |
| 7 | \( 1 + (-0.642 - 0.766i)T \) |
| 13 | \( 1 - iT \) |
good | 3 | \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1.32 - 0.766i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.684iT - T^{2} \) |
| 47 | \( 1 + (0.300 - 0.173i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.96T + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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
−8.994323986615914114054147566543, −8.263529233258500339511769490362, −6.97860813712941421047016158133, −6.31233048908912313491532764695, −5.18687718668973217609122716531, −4.80387325523212966418954571897, −4.11276245204479178983258217529, −3.07157360548803723127928548877, −2.25228183856220593164197670024, −1.66561102296930731723405124422,
1.93547240257327305875958737663, 2.27891096381898953825989990816, 3.40792952469189647150167663179, 3.95592340749042479672300415738, 5.07531866955763064060284918027, 5.88740993437270763709438244533, 6.73660586762038659457370753704, 7.25315066942754115172321832355, 7.88697033069473447411413452840, 8.475882486274791064362554802015