L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + (0.866 − 0.499i)6-s + i·7-s − 8-s + (−0.866 + 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)14-s + (−0.499 − 0.866i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − i·19-s + (0.5 − 0.866i)21-s + (−0.5 − 0.866i)23-s + (0.866 + 0.499i)24-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + (0.866 − 0.499i)6-s + i·7-s − 8-s + (−0.866 + 0.499i)10-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)14-s + (−0.499 − 0.866i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − i·19-s + (0.5 − 0.866i)21-s + (−0.5 − 0.866i)23-s + (0.866 + 0.499i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4405601059\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4405601059\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - iT \) |
| 19 | \( 1 + iT \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 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 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 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
−13.72272730390624963018605645142, −12.58533157210974718606194468804, −11.74517020196033499914094634297, −10.71499554360821088389618511129, −9.239053692940039465874360737458, −8.456438373398409426979478515721, −6.86187592916030067191584686767, −6.30950169152618461941082530168, −5.43296173482710314264454112292, −2.71044367256582579166473223063,
1.69683603271289636518353803388, 3.99891080945266620293898225975, 5.53890719816988880382030444731, 6.43424301154797598532413666283, 8.385138826224672845299114718856, 9.656034681280208466440361484716, 10.28944621613027588315661346404, 11.08243332735907686790328381905, 11.87241087780795391720983449716, 13.23459958368733222205607303095