L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.266 + 0.223i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)6-s + (0.5 + 0.866i)8-s + (−0.152 + 0.866i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.300i)12-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + 0.879·18-s + (−0.939 + 0.342i)19-s + (−1.17 + 0.984i)22-s + (−0.326 − 0.118i)24-s + (−0.326 − 0.565i)27-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.266 + 0.223i)3-s + (−0.939 + 0.342i)4-s + (0.266 + 0.223i)6-s + (0.5 + 0.866i)8-s + (−0.152 + 0.866i)9-s + (−0.766 − 1.32i)11-s + (0.173 − 0.300i)12-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + 0.879·18-s + (−0.939 + 0.342i)19-s + (−1.17 + 0.984i)22-s + (−0.326 − 0.118i)24-s + (−0.326 − 0.565i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2409417154\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2409417154\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (0.939 - 0.342i)T \) |
good | 3 | \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 83 | \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
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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
−8.755768467589429968913737265955, −8.270103621624143949922350970897, −7.88512327069241846971942814854, −6.55353230944259758682397581768, −5.63405738128980571205764579543, −5.09920629513079790642830955535, −4.16672468280027395878170425172, −3.36067603196106313675791407586, −2.50540148636659873801102332292, −1.51538650696186285296774844298,
0.14961762086333696360935269126, 1.69981268090546034629691310302, 3.01600727090980773279987997249, 4.14867046600230301750422647157, 4.89030433668280668522632912268, 5.52060509807175794276420049309, 6.45588296885371684929170856879, 6.99469118156736561098968252179, 7.52042483320470657897726672305, 8.430288526080192764186721374742