L(s) = 1 | + (1.5 + 2.59i)5-s − 4·11-s + (2.5 − 4.33i)13-s + (−2.5 − 4.33i)17-s + (−4 − 1.73i)19-s + (0.5 − 0.866i)23-s + (−2 + 3.46i)25-s + (1.5 − 2.59i)29-s − 4·31-s + 2·37-s + (−2.5 − 4.33i)41-s + (−5.5 − 9.52i)43-s + (2.5 − 4.33i)47-s − 7·49-s + (−4.5 + 7.79i)53-s + ⋯ |
L(s) = 1 | + (0.670 + 1.16i)5-s − 1.20·11-s + (0.693 − 1.20i)13-s + (−0.606 − 1.05i)17-s + (−0.917 − 0.397i)19-s + (0.104 − 0.180i)23-s + (−0.400 + 0.692i)25-s + (0.278 − 0.482i)29-s − 0.718·31-s + 0.328·37-s + (−0.390 − 0.676i)41-s + (−0.838 − 1.45i)43-s + (0.364 − 0.631i)47-s − 49-s + (−0.618 + 1.07i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.019907464\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.019907464\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.5 + 4.33i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.5 + 11.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.5 - 7.79i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.5 - 14.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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
−8.546191163925563324210522558179, −7.85437099243471296079175975376, −7.00134717004277172353344143879, −6.40253052260283404219307950030, −5.56117601873884063399741127627, −4.88939014141834088097914918072, −3.57740626301042166320578420215, −2.75225708006681505704631211792, −2.14300460157535248962369625169, −0.30614336638079191187775652499,
1.43697213796588012672250153321, 2.08506452467576889631512769217, 3.45133329499950360906809124492, 4.54993378919572152849517390832, 4.97480591990136706566858905130, 6.09357784928277278171943630680, 6.42799176659838320168571650820, 7.74318730732949618549262307842, 8.370674379851304223787298597149, 9.003434351026882252085404861099