L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.382 + 0.923i)3-s + (−0.866 − 0.499i)4-s + (−0.991 + 0.130i)5-s + (−0.793 − 0.608i)6-s + (−0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.130 − 0.991i)10-s + (0.793 − 0.608i)12-s + (1.78 − 0.478i)13-s + (0.499 − 0.866i)14-s + (0.258 − 0.965i)15-s + (0.500 + 0.866i)16-s + (0.866 − 0.5i)18-s + 1.21i·19-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.382 + 0.923i)3-s + (−0.866 − 0.499i)4-s + (−0.991 + 0.130i)5-s + (−0.793 − 0.608i)6-s + (−0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.130 − 0.991i)10-s + (0.793 − 0.608i)12-s + (1.78 − 0.478i)13-s + (0.499 − 0.866i)14-s + (0.258 − 0.965i)15-s + (0.500 + 0.866i)16-s + (0.866 − 0.5i)18-s + 1.21i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0136 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0136 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5783485311\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5783485311\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 3 | \( 1 + (0.382 - 0.923i)T \) |
| 5 | \( 1 + (0.991 - 0.130i)T \) |
| 7 | \( 1 + (0.965 + 0.258i)T \) |
good | 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.78 + 0.478i)T + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - 1.21iT - T^{2} \) |
| 23 | \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.793 - 1.37i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - 0.517iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.739 - 0.198i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
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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
−9.041703626739663900274692860129, −8.526696751430362994412894998006, −7.912495230600610778596344762357, −6.81081688828747081981406739023, −6.20427401976611485610364991021, −5.64593637607992115662820334593, −4.40176075454693580079574903700, −3.91698780365689999440888362944, −3.20329923843205659276944978048, −0.73256605985713212924672187335,
0.78419546430315169502782000294, 1.94333846474339680527151085424, 3.21996817578532277176595914503, 3.69688172125175613250799598663, 4.85097595113090488894646136321, 5.84712743320767800443947595174, 6.70444988136390938911195471725, 7.49455064213313763004489593468, 8.258769839440015503101687933763, 8.903287490212388380681590149295