L(s) = 1 | − 7-s − 2·11-s + (−2.5 + 4.33i)13-s + (−2 − 3.46i)17-s + (4 + 1.73i)19-s + (2 − 3.46i)23-s + (2.5 − 4.33i)25-s + (−4 + 6.92i)29-s + 3·31-s + 3·37-s + (−6 − 10.3i)41-s + (−0.5 − 0.866i)43-s + (3 − 5.19i)47-s − 6·49-s + (2 − 3.46i)53-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 0.603·11-s + (−0.693 + 1.20i)13-s + (−0.485 − 0.840i)17-s + (0.917 + 0.397i)19-s + (0.417 − 0.722i)23-s + (0.5 − 0.866i)25-s + (−0.742 + 1.28i)29-s + 0.538·31-s + 0.493·37-s + (−0.937 − 1.62i)41-s + (−0.0762 − 0.132i)43-s + (0.437 − 0.757i)47-s − 0.857·49-s + (0.274 − 0.475i)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
\(L(1)\) |
\(\approx\) |
\(0.9505928514\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9505928514\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4 - 6.92i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + (6 + 10.3i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.702626834706151081135808636067, −7.84916955288911936803394069730, −6.87972689111588910568510461605, −6.66949586248055424210420187394, −5.29652923397125957659093818458, −4.87183394052702909708307565178, −3.78632877542936946812539949477, −2.81957979243773189861096770790, −1.93055082428825596053398305041, −0.32873072987199713083536558795,
1.14091470839979226297207057656, 2.60558213526454230453056630330, 3.19947756374256917372779807852, 4.32737447084726215117584665988, 5.26194273546154851020334709692, 5.82593216827826283901988619819, 6.80599851937037139546779841251, 7.64148185930982868331252728535, 8.082392866029877861500992519325, 9.124362721225561214403692272046