L(s) = 1 | + (0.438 − 0.898i)3-s + (0.961 − 0.275i)4-s + (0.704 + 1.74i)5-s + (−0.615 − 0.788i)9-s + (0.173 − 0.984i)12-s + (1.87 + 0.131i)15-s + (0.848 − 0.529i)16-s + (1.15 + 1.48i)20-s + (0.939 + 0.342i)23-s + (−1.82 + 1.75i)25-s + (−0.978 + 0.207i)27-s + (0.0121 − 0.347i)31-s + (−0.809 − 0.587i)36-s + (−1.71 − 0.764i)37-s + (0.939 − 1.62i)45-s + ⋯ |
L(s) = 1 | + (0.438 − 0.898i)3-s + (0.961 − 0.275i)4-s + (0.704 + 1.74i)5-s + (−0.615 − 0.788i)9-s + (0.173 − 0.984i)12-s + (1.87 + 0.131i)15-s + (0.848 − 0.529i)16-s + (1.15 + 1.48i)20-s + (0.939 + 0.342i)23-s + (−1.82 + 1.75i)25-s + (−0.978 + 0.207i)27-s + (0.0121 − 0.347i)31-s + (−0.809 − 0.587i)36-s + (−1.71 − 0.764i)37-s + (0.939 − 1.62i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.061491441\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.061491441\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.438 + 0.898i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.961 + 0.275i)T^{2} \) |
| 5 | \( 1 + (-0.704 - 1.74i)T + (-0.719 + 0.694i)T^{2} \) |
| 7 | \( 1 + (0.374 + 0.927i)T^{2} \) |
| 13 | \( 1 + (-0.559 + 0.829i)T^{2} \) |
| 17 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 19 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.615 - 0.788i)T^{2} \) |
| 31 | \( 1 + (-0.0121 + 0.347i)T + (-0.997 - 0.0697i)T^{2} \) |
| 37 | \( 1 + (1.71 + 0.764i)T + (0.669 + 0.743i)T^{2} \) |
| 41 | \( 1 + (0.615 + 0.788i)T^{2} \) |
| 43 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-1.47 - 0.422i)T + (0.848 + 0.529i)T^{2} \) |
| 53 | \( 1 + (-0.107 - 0.330i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (0.0840 - 0.336i)T + (-0.882 - 0.469i)T^{2} \) |
| 61 | \( 1 + (0.997 - 0.0697i)T^{2} \) |
| 67 | \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (1.49 - 0.318i)T + (0.913 - 0.406i)T^{2} \) |
| 73 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 79 | \( 1 + (-0.961 + 0.275i)T^{2} \) |
| 83 | \( 1 + (-0.559 - 0.829i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.573 - 1.42i)T + (-0.719 - 0.694i)T^{2} \) |
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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
−8.794670528503894229608786733961, −7.66421841468708950137810713214, −7.14682856603976762981306261175, −6.78521339239600798282900029607, −6.00360172394265549348963061573, −5.48811383377481780904857637605, −3.64799283549902490086836126624, −2.92823833144669977402999159120, −2.33128946896980655562396356309, −1.52313331061645885916056321115,
1.37042470119783972349165009494, 2.29804706101768272394118859647, 3.27920695354294147415523577543, 4.26921181497280876039410201966, 5.04996032944503141321571799162, 5.58702077493648664175377619135, 6.49959737211994574743338404599, 7.53996800540065989219592396443, 8.362022848573165915572328709557, 8.830229065042818446435898011001