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 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.65176902643979957407562522284, −9.611260232005024940752460231106, −8.952138043441643345854234310768, −7.933657381882974717275263284623, −6.94265622070160731094303134763, −6.30556778419459815176130522348, −5.05780718076940847184524796367, −3.96870431994406708542770645362, −2.98585148995980340121253393424, −1.37326365594736512890603353636,
0.977606918264635694411434422851, 2.59662526913971519372896036577, 4.02060887575246387022468094790, 4.68464939759918247971089997218, 6.12993635719395679725819795471, 6.65806425345277821124763210627, 8.015615020936770040446543523647, 8.580652816347476968760742032571, 9.497351295310734943671927488449, 10.39055879973203081217386114195