L(s) = 1 | + (0.615 + 0.788i)3-s + (0.848 − 0.529i)4-s + (0.893 + 0.924i)5-s + (−0.241 + 0.970i)9-s + (0.939 + 0.342i)12-s + (−0.178 + 1.27i)15-s + (0.438 − 0.898i)16-s + (1.24 + 0.311i)20-s + (1.11 − 1.32i)23-s + (−0.0227 + 0.652i)25-s + (−0.913 + 0.406i)27-s + (−1.87 − 0.131i)31-s + (0.309 + 0.951i)36-s + (−1.02 − 1.13i)37-s + (−1.11 + 0.642i)45-s + ⋯ |
L(s) = 1 | + (0.615 + 0.788i)3-s + (0.848 − 0.529i)4-s + (0.893 + 0.924i)5-s + (−0.241 + 0.970i)9-s + (0.939 + 0.342i)12-s + (−0.178 + 1.27i)15-s + (0.438 − 0.898i)16-s + (1.24 + 0.311i)20-s + (1.11 − 1.32i)23-s + (−0.0227 + 0.652i)25-s + (−0.913 + 0.406i)27-s + (−1.87 − 0.131i)31-s + (0.309 + 0.951i)36-s + (−1.02 − 1.13i)37-s + (−1.11 + 0.642i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.199330915\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.199330915\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.615 - 0.788i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.848 + 0.529i)T^{2} \) |
| 5 | \( 1 + (-0.893 - 0.924i)T + (-0.0348 + 0.999i)T^{2} \) |
| 7 | \( 1 + (-0.719 + 0.694i)T^{2} \) |
| 13 | \( 1 + (-0.374 - 0.927i)T^{2} \) |
| 17 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 19 | \( 1 + (-0.104 - 0.994i)T^{2} \) |
| 23 | \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (0.241 + 0.970i)T^{2} \) |
| 31 | \( 1 + (1.87 + 0.131i)T + (0.990 + 0.139i)T^{2} \) |
| 37 | \( 1 + (1.02 + 1.13i)T + (-0.104 + 0.994i)T^{2} \) |
| 41 | \( 1 + (0.241 - 0.970i)T^{2} \) |
| 43 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (1.04 - 1.67i)T + (-0.438 - 0.898i)T^{2} \) |
| 53 | \( 1 + (-0.402 - 0.553i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.321 + 0.603i)T + (-0.559 - 0.829i)T^{2} \) |
| 61 | \( 1 + (0.990 - 0.139i)T^{2} \) |
| 67 | \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.801 - 1.79i)T + (-0.669 + 0.743i)T^{2} \) |
| 73 | \( 1 + (0.913 - 0.406i)T^{2} \) |
| 79 | \( 1 + (0.848 - 0.529i)T^{2} \) |
| 83 | \( 1 + (0.374 - 0.927i)T^{2} \) |
| 89 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.249 + 0.241i)T + (0.0348 + 0.999i)T^{2} \) |
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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
−9.153689256151153156793864475808, −8.277268301569945820397678054093, −7.18062771173498819076120597638, −6.84531076267642127780004506138, −5.79037361204289476849971649785, −5.34404990034968275656173051254, −4.20860297969104187158885032631, −3.10173019682214830690256068898, −2.54503739463375259760213341868, −1.73029750469992573987548461156,
1.43708086815946960770463360257, 1.91050168910141843602122568696, 3.03602366620213668387788919251, 3.71174777594770913522595220722, 5.12345771744203805805062705191, 5.74931483869707306727271614993, 6.66701798595500699876806295497, 7.24122903127454778885106112968, 7.916425027549621932970832500979, 8.811571150569839372923550699156