L(s) = 1 | + i·3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s − 9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + (0.866 − 0.5i)15-s + i·19-s + (0.866 + 0.5i)21-s + (−0.5 − 0.866i)23-s − i·27-s + (0.866 + 0.5i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s − 0.999·35-s + ⋯ |
L(s) = 1 | + i·3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s − 9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + (0.866 − 0.5i)15-s + i·19-s + (0.866 + 0.5i)21-s + (−0.5 − 0.866i)23-s − i·27-s + (0.866 + 0.5i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s − 0.999·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.128399057\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128399057\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 19 | \( 1 - iT \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.880399765641498956628931968453, −8.205078541238790388329651773956, −7.85787638298576248819930082371, −6.68836878897320700480019207427, −5.65111771491815810474994982653, −4.90894974542624517736927906414, −4.21195935534359479455530414920, −3.75510182771274784213227387490, −2.40186543378925588299397966844, −0.866814079244941949104423492544,
1.21263607528423496615475865281, 2.56575915008814794383861386600, 2.99401969470734593869664762347, 4.21885901724040966352168439358, 5.45144004586959029827187297512, 6.04063760208907748150002656685, 6.78928803717125547005704235005, 7.50021725428402051919473591594, 8.345217973022448612560413197203, 8.625689496750483960989559394930