L(s) = 1 | + (1.40 − 0.147i)2-s + (0.978 − 0.207i)4-s + (−0.104 + 0.994i)5-s + (−1.05 + 0.946i)7-s + 1.41i·10-s + (−1.33 + 1.48i)14-s + (−0.913 + 0.406i)16-s + (0.104 + 0.994i)20-s + (−0.831 + 1.14i)28-s + (−1.05 + 0.946i)29-s + (0.913 + 0.406i)31-s + (−1.22 + 0.707i)32-s + (−0.831 − 1.14i)35-s + (−0.309 + 0.951i)37-s + (−0.978 − 0.207i)47-s + ⋯ |
L(s) = 1 | + (1.40 − 0.147i)2-s + (0.978 − 0.207i)4-s + (−0.104 + 0.994i)5-s + (−1.05 + 0.946i)7-s + 1.41i·10-s + (−1.33 + 1.48i)14-s + (−0.913 + 0.406i)16-s + (0.104 + 0.994i)20-s + (−0.831 + 1.14i)28-s + (−1.05 + 0.946i)29-s + (0.913 + 0.406i)31-s + (−1.22 + 0.707i)32-s + (−0.831 − 1.14i)35-s + (−0.309 + 0.951i)37-s + (−0.978 − 0.207i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.863043163\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.863043163\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.40 + 0.147i)T + (0.978 - 0.207i)T^{2} \) |
| 5 | \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 7 | \( 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2} \) |
| 13 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2} \) |
| 31 | \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 53 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \) |
| 61 | \( 1 + (-0.575 - 1.29i)T + (-0.669 + 0.743i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 83 | \( 1 + (0.575 + 1.29i)T + (-0.669 + 0.743i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
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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
−9.026775839333196717395495560932, −8.341076730281621821946585216526, −6.94986669738967590067744759258, −6.75147577689689344433273353473, −5.84770657118862845938142528685, −5.33269479648490522544252580214, −4.31980219003254571432734812613, −3.26218971121575888348562575499, −3.06566255256318579287432348131, −2.09398522342391422671063326131,
0.66363393707226686707493003493, 2.32212971261426102653017932273, 3.46719624172469146832711879784, 3.97860295629957547862195025534, 4.69712282445656984679934571940, 5.45368702477807064557538525819, 6.22680745287204869794609895919, 6.86364901675603511333815398255, 7.66449787655207837804233516704, 8.593854736033989733762383155611