L(s) = 1 | + (−0.5 − 0.866i)5-s + 4·11-s + (0.5 − 0.866i)13-s + (1.5 + 2.59i)17-s + (−4 − 1.73i)19-s + (2.5 − 4.33i)23-s + (2 − 3.46i)25-s + (3.5 − 6.06i)29-s + 4·31-s + 10·37-s + (−2.5 − 4.33i)41-s + (2.5 + 4.33i)43-s + (−3.5 + 6.06i)47-s − 7·49-s + (5.5 − 9.52i)53-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + 1.20·11-s + (0.138 − 0.240i)13-s + (0.363 + 0.630i)17-s + (−0.917 − 0.397i)19-s + (0.521 − 0.902i)23-s + (0.400 − 0.692i)25-s + (0.649 − 1.12i)29-s + 0.718·31-s + 1.64·37-s + (−0.390 − 0.676i)41-s + (0.381 + 0.660i)43-s + (−0.510 + 0.884i)47-s − 49-s + (0.755 − 1.30i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45338 - 0.466753i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45338 - 0.466753i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.5 + 4.33i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.5 - 9.52i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.5 + 12.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.39055879973203081217386114195, −9.497351295310734943671927488449, −8.580652816347476968760742032571, −8.015615020936770040446543523647, −6.65806425345277821124763210627, −6.12993635719395679725819795471, −4.68464939759918247971089997218, −4.02060887575246387022468094790, −2.59662526913971519372896036577, −0.977606918264635694411434422851,
1.37326365594736512890603353636, 2.98585148995980340121253393424, 3.96870431994406708542770645362, 5.05780718076940847184524796367, 6.30556778419459815176130522348, 6.94265622070160731094303134763, 7.933657381882974717275263284623, 8.952138043441643345854234310768, 9.611260232005024940752460231106, 10.65176902643979957407562522284