L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 2·5-s + 0.999·8-s + (−1 + 1.73i)10-s − 11-s + (−3 + 5.19i)13-s + (−0.5 + 0.866i)16-s + (2.5 − 4.33i)17-s + (−3.5 − 6.06i)19-s + (−0.999 − 1.73i)20-s + (0.5 − 0.866i)22-s − 4·23-s − 25-s + (−3 − 5.19i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.894·5-s + 0.353·8-s + (−0.316 + 0.547i)10-s − 0.301·11-s + (−0.832 + 1.44i)13-s + (−0.125 + 0.216i)16-s + (0.606 − 1.05i)17-s + (−0.802 − 1.39i)19-s + (−0.223 − 0.387i)20-s + (0.106 − 0.184i)22-s − 0.834·23-s − 0.200·25-s + (−0.588 − 1.01i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6395124594\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6395124594\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6 - 10.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8 + 13.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 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
−8.810661984264330921149051843940, −7.76311535284028798622245006652, −7.13271091214475350525134201749, −6.43617715905182165911512521095, −5.66489699492468480610430555085, −4.88188944588736879478920717869, −4.11382886085275281724842610614, −2.55245853285665093412408415442, −1.88657746326511693864336009756, −0.21945234658843250227167295579,
1.43636455557231033643852398841, 2.23955877835577022395845030529, 3.25544249892602445848897696443, 4.09074320107573478716896028619, 5.43887385743767058580437677234, 5.68649778031052664928837546195, 6.80253076925883842234073445890, 7.937426045493439712861710692592, 8.168340322679005512462088057496, 9.217637976670619819793185243771