L(s) = 1 | + (−2.86 + 1.65i)3-s + (0.877 + 2.05i)5-s + (−0.517 + 0.895i)7-s + (3.96 − 6.86i)9-s + (−2.96 + 1.70i)11-s + (−3.57 − 0.455i)13-s + (−5.91 − 4.43i)15-s + (−2.07 − 1.19i)17-s + (−5.37 − 3.10i)19-s − 3.41i·21-s + (6.28 − 3.62i)23-s + (−3.46 + 3.60i)25-s + 16.3i·27-s + (0.902 + 1.56i)29-s + 5.80i·31-s + ⋯ |
L(s) = 1 | + (−1.65 + 0.954i)3-s + (0.392 + 0.919i)5-s + (−0.195 + 0.338i)7-s + (1.32 − 2.28i)9-s + (−0.892 + 0.515i)11-s + (−0.992 − 0.126i)13-s + (−1.52 − 1.14i)15-s + (−0.503 − 0.290i)17-s + (−1.23 − 0.711i)19-s − 0.746i·21-s + (1.31 − 0.756i)23-s + (−0.692 + 0.721i)25-s + 3.13i·27-s + (0.167 + 0.290i)29-s + 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0422629 - 0.318579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0422629 - 0.318579i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.877 - 2.05i)T \) |
| 13 | \( 1 + (3.57 + 0.455i)T \) |
good | 3 | \( 1 + (2.86 - 1.65i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (0.517 - 0.895i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.96 - 1.70i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.07 + 1.19i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.37 + 3.10i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.28 + 3.62i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.902 - 1.56i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.80iT - 31T^{2} \) |
| 37 | \( 1 + (-0.713 - 1.23i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.60 - 2.07i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.86 + 1.07i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.50T + 47T^{2} \) |
| 53 | \( 1 - 4.55iT - 53T^{2} \) |
| 59 | \( 1 + (5.06 + 2.92i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.90 - 3.29i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.80 - 6.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.49 - 5.48i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 7.15T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 0.706T + 83T^{2} \) |
| 89 | \( 1 + (-5.06 + 2.92i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.99 - 13.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.34483223232978437372253442048, −11.27628034027716776452035590017, −10.60090875615577188477239826894, −10.06950745968034242481675831609, −9.053693597758349805491400939141, −7.07388070907955516355875027971, −6.47206456109769516222321192762, −5.27007665067552371159395266131, −4.58192183504811074584348513649, −2.78261673492602890302361159346,
0.29018466030807157641793426373, 1.93928100567016866983156915873, 4.58830673279857770953107287101, 5.43183757781517764060190807795, 6.28511735943395282409189902189, 7.33257584168053171486773271563, 8.325257069396258376543715699713, 9.848721658482037881586488540186, 10.71994789550245837000059464419, 11.56485985055949506071206974960