L(s) = 1 | + (1 + 1.73i)5-s + (0.5 + 2.59i)7-s − 2·13-s + (−3 + 5.19i)17-s + (0.5 + 0.866i)19-s + (1 + 1.73i)23-s + (0.500 − 0.866i)25-s − 6·29-s + (−0.5 + 0.866i)31-s + (−4 + 3.46i)35-s + (−1 − 1.73i)37-s − 2·41-s + 9·43-s + (−1 − 1.73i)47-s + (−6.5 + 2.59i)49-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (0.188 + 0.981i)7-s − 0.554·13-s + (−0.727 + 1.26i)17-s + (0.114 + 0.198i)19-s + (0.208 + 0.361i)23-s + (0.100 − 0.173i)25-s − 1.11·29-s + (−0.0898 + 0.155i)31-s + (−0.676 + 0.585i)35-s + (−0.164 − 0.284i)37-s − 0.312·41-s + 1.37·43-s + (−0.145 − 0.252i)47-s + (−0.928 + 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.362017775\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.362017775\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - T + 97T^{2} \) |
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
−9.644093885762867464957976406461, −9.066760792709629262092133775669, −8.177643758055641779014233342998, −7.33191668927270503132816082471, −6.36613732232801086418716228722, −5.81585193295113263537956030565, −4.85780768877808794439349875568, −3.68952999965831902844520463433, −2.59438019581345098769011813801, −1.82580531957741925980449126311,
0.51286791491399577784254189146, 1.78463812972302615995385256737, 3.04248509884200060114048636225, 4.33671913434408911290457889828, 4.89099901423239278384706201682, 5.79193808526861012654863782863, 6.98992240258271911981076863406, 7.42418892856606446129384482202, 8.486844716697131649978032267011, 9.305199930688179908099916108076