L(s) = 1 | + 1.03·5-s + (1.07 − 2.41i)7-s − 1.58·11-s + (−2.52 + 4.37i)13-s + (−2.58 + 4.47i)17-s + (−0.392 − 0.680i)19-s − 5.86·23-s − 3.93·25-s + (4.44 + 7.69i)29-s + (0.575 + 0.996i)31-s + (1.10 − 2.49i)35-s + (4.07 + 7.06i)37-s + (−3.87 + 6.70i)41-s + (1.26 + 2.19i)43-s + (−4.24 + 7.35i)47-s + ⋯ |
L(s) = 1 | + 0.461·5-s + (0.406 − 0.913i)7-s − 0.478·11-s + (−0.700 + 1.21i)13-s + (−0.626 + 1.08i)17-s + (−0.0901 − 0.156i)19-s − 1.22·23-s − 0.787·25-s + (0.825 + 1.42i)29-s + (0.103 + 0.179i)31-s + (0.187 − 0.421i)35-s + (0.670 + 1.16i)37-s + (−0.604 + 1.04i)41-s + (0.193 + 0.334i)43-s + (−0.619 + 1.07i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.114774033\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.114774033\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.07 + 2.41i)T \) |
good | 5 | \( 1 - 1.03T + 5T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 + (2.52 - 4.37i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.58 - 4.47i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.392 + 0.680i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 + (-4.44 - 7.69i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.575 - 0.996i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.07 - 7.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.87 - 6.70i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.26 - 2.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.24 - 7.35i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.41 + 4.17i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.93 + 3.35i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.82 + 8.35i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.837 - 1.45i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + (-3.04 + 5.27i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.15 + 7.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.12 - 12.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.69 - 11.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.67 + 4.63i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.420389147654390836619030827475, −8.247852391141230818083219528840, −7.898348089469415978506940438054, −6.66400183266040526387830507267, −6.44138162934173970926320511911, −5.09550694393416918582523481989, −4.50052836778015979704343453262, −3.62348142183726107718253868316, −2.28940109952868211874987040623, −1.46107567666792661649050290385,
0.35886743517351276645601851471, 2.22926191744209476519025877888, 2.54165243900902740189645437324, 3.94727506117385583571141894819, 5.03439118063053629714279591747, 5.58114292417273048179007346747, 6.27821586412644776334203622392, 7.44607935829195695711973738733, 8.018749612846578222324452720084, 8.761706666192673492011973136927