L(s) = 1 | + (0.939 + 0.342i)2-s + (−0.0603 − 0.342i)3-s + (0.766 + 0.642i)4-s + (0.0603 − 0.342i)6-s + (0.500 + 0.866i)8-s + (0.826 − 0.300i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.300i)12-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + 0.879·18-s + (−0.266 − 1.50i)22-s + (0.266 − 0.223i)24-s + (0.173 − 0.984i)25-s + (−0.326 − 0.565i)27-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)2-s + (−0.0603 − 0.342i)3-s + (0.766 + 0.642i)4-s + (0.0603 − 0.342i)6-s + (0.500 + 0.866i)8-s + (0.826 − 0.300i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.300i)12-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + 0.879·18-s + (−0.266 − 1.50i)22-s + (0.266 − 0.223i)24-s + (0.173 − 0.984i)25-s + (−0.326 − 0.565i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.302551026\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.302551026\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \) |
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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
−8.568421291310552712805491210612, −8.067461537656153004816954777061, −7.38365929111043142930653691619, −6.48835563295518069368913904480, −5.97501440763754502434756709736, −5.16546701928589593357337607545, −4.28298592218771649964360424099, −3.39827781552264470340382103731, −2.64014754148139581528221999923, −1.29867225161554197780019088560,
1.51114171700828586334305718709, 2.40269051069389801021545738612, 3.48260087559128913277403603752, 4.25536216984020032170730764461, 5.11109289041087636757503069193, 5.42965869265291008057603608622, 6.70761662365950432500731436330, 7.31797326962364547845153638051, 7.86833770032906200584188759496, 9.327840505586410548383037360318