L(s) = 1 | + (−0.410 − 0.315i)3-s + (0.965 − 0.258i)4-s + (0.665 − 1.34i)5-s + (−0.189 − 0.707i)9-s + (−0.130 + 0.991i)11-s + (−0.478 − 0.198i)12-s + (−0.698 + 0.344i)15-s + (0.866 − 0.499i)16-s + (0.293 − 1.47i)20-s + (−1.18 + 1.34i)23-s + (−0.767 − 1.00i)25-s + (−0.343 + 0.828i)27-s + (−0.465 + 0.607i)31-s + (0.366 − 0.366i)33-s + (−0.366 − 0.633i)36-s + (0.0983 − 0.0862i)37-s + ⋯ |
L(s) = 1 | + (−0.410 − 0.315i)3-s + (0.965 − 0.258i)4-s + (0.665 − 1.34i)5-s + (−0.189 − 0.707i)9-s + (−0.130 + 0.991i)11-s + (−0.478 − 0.198i)12-s + (−0.698 + 0.344i)15-s + (0.866 − 0.499i)16-s + (0.293 − 1.47i)20-s + (−1.18 + 1.34i)23-s + (−0.767 − 1.00i)25-s + (−0.343 + 0.828i)27-s + (−0.465 + 0.607i)31-s + (0.366 − 0.366i)33-s + (−0.366 − 0.633i)36-s + (0.0983 − 0.0862i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.204917232\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.204917232\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.130 - 0.991i)T \) |
| 97 | \( 1 + (0.991 - 0.130i)T \) |
good | 2 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 3 | \( 1 + (0.410 + 0.315i)T + (0.258 + 0.965i)T^{2} \) |
| 5 | \( 1 + (-0.665 + 1.34i)T + (-0.608 - 0.793i)T^{2} \) |
| 7 | \( 1 + (-0.991 - 0.130i)T^{2} \) |
| 13 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 17 | \( 1 + (0.991 - 0.130i)T^{2} \) |
| 19 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 23 | \( 1 + (1.18 - 1.34i)T + (-0.130 - 0.991i)T^{2} \) |
| 29 | \( 1 + (0.793 - 0.608i)T^{2} \) |
| 31 | \( 1 + (0.465 - 0.607i)T + (-0.258 - 0.965i)T^{2} \) |
| 37 | \( 1 + (-0.0983 + 0.0862i)T + (0.130 - 0.991i)T^{2} \) |
| 41 | \( 1 + (-0.793 + 0.608i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 53 | \( 1 + (-0.158 - 1.20i)T + (-0.965 + 0.258i)T^{2} \) |
| 59 | \( 1 + (0.869 + 0.991i)T + (-0.130 + 0.991i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.57 + 1.05i)T + (0.382 - 0.923i)T^{2} \) |
| 71 | \( 1 + (-1.88 - 0.641i)T + (0.793 + 0.608i)T^{2} \) |
| 73 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (-0.991 + 0.130i)T^{2} \) |
| 89 | \( 1 + (0.707 - 0.707i)T^{2} \) |
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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
−9.715888251913268029963682549944, −9.393637586347149914189366873502, −8.222021181446643724287478507300, −7.31712962202594049896171584747, −6.45951665967265853564704726601, −5.65979440009865465628602533797, −5.05842419301597867417898419688, −3.70758133417618617821232739416, −2.12547522138973401958783025083, −1.28629528881719978592292595901,
2.15701465607510133754016963113, 2.79402747935443441209340765618, 3.90997870486005752011527872466, 5.42698506389916229780727067058, 6.14703699773596388431557928923, 6.68612427429077593130445086102, 7.71354387435317689300307617706, 8.410437949910000476942818472890, 9.834666142574791018442992719543, 10.45380001371770723612163586070