L(s) = 1 | + (−0.374 + 0.927i)3-s + (−0.882 − 0.469i)4-s + (0.333 + 0.0957i)5-s + (−0.719 − 0.694i)9-s + (0.766 − 0.642i)12-s + (−0.213 + 0.273i)15-s + (0.559 + 0.829i)16-s + (−0.249 − 0.241i)20-s + (−0.173 + 0.984i)23-s + (−0.745 − 0.466i)25-s + (0.913 − 0.406i)27-s + (0.671 + 1.37i)31-s + (0.309 + 0.951i)36-s + (0.232 + 0.258i)37-s + (−0.173 − 0.300i)45-s + ⋯ |
L(s) = 1 | + (−0.374 + 0.927i)3-s + (−0.882 − 0.469i)4-s + (0.333 + 0.0957i)5-s + (−0.719 − 0.694i)9-s + (0.766 − 0.642i)12-s + (−0.213 + 0.273i)15-s + (0.559 + 0.829i)16-s + (−0.249 − 0.241i)20-s + (−0.173 + 0.984i)23-s + (−0.745 − 0.466i)25-s + (0.913 − 0.406i)27-s + (0.671 + 1.37i)31-s + (0.309 + 0.951i)36-s + (0.232 + 0.258i)37-s + (−0.173 − 0.300i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7560046310\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7560046310\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.374 - 0.927i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.882 + 0.469i)T^{2} \) |
| 5 | \( 1 + (-0.333 - 0.0957i)T + (0.848 + 0.529i)T^{2} \) |
| 7 | \( 1 + (-0.961 - 0.275i)T^{2} \) |
| 13 | \( 1 + (-0.990 - 0.139i)T^{2} \) |
| 17 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 19 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 23 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (0.719 - 0.694i)T^{2} \) |
| 31 | \( 1 + (-0.671 - 1.37i)T + (-0.615 + 0.788i)T^{2} \) |
| 37 | \( 1 + (-0.232 - 0.258i)T + (-0.104 + 0.994i)T^{2} \) |
| 41 | \( 1 + (0.719 + 0.694i)T^{2} \) |
| 43 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 47 | \( 1 + (-1.65 + 0.882i)T + (0.559 - 0.829i)T^{2} \) |
| 53 | \( 1 + (1.23 - 0.900i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.0534 - 1.53i)T + (-0.997 + 0.0697i)T^{2} \) |
| 61 | \( 1 + (0.615 + 0.788i)T^{2} \) |
| 67 | \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (1.71 - 0.764i)T + (0.669 - 0.743i)T^{2} \) |
| 73 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 79 | \( 1 + (0.882 + 0.469i)T^{2} \) |
| 83 | \( 1 + (-0.990 + 0.139i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.80 - 0.518i)T + (0.848 - 0.529i)T^{2} \) |
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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
−9.050960136289366419593745938357, −8.637158030728076315017741389300, −7.62151081508282466530704420395, −6.51665097578850996416212719698, −5.73680649814235703277274320778, −5.29506522032021903731567946071, −4.37246417972263713973267633988, −3.81607552773239057140214304109, −2.70902136989793618229298369817, −1.18804230737929630620371305895,
0.55940335244794128740168162423, 1.94613481840525231356875573787, 2.90644044279521253690512904335, 4.04601829995605203434967643702, 4.83838935009889222380741696731, 5.70401542022193622890351206143, 6.29023150315534666351779658869, 7.28026871162704384566345343401, 7.903223454764158464403172269909, 8.480100029152596744588295178462