L(s) = 1 | + (−0.5 − 0.866i)2-s + i·3-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s + 7-s − 8-s − 9-s + (0.866 + 0.499i)10-s + (0.5 + 0.866i)11-s + i·13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.5 + 0.866i)18-s + (−0.866 + 0.5i)19-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + i·3-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s + 7-s − 8-s − 9-s + (0.866 + 0.499i)10-s + (0.5 + 0.866i)11-s + i·13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.5 + 0.866i)18-s + (−0.866 + 0.5i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6878120750\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6878120750\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + 2iT - T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.53854399045989399516714828311, −9.884899331427009290046113166777, −9.051853996844306583921864469087, −8.295011829656140946017713180545, −7.23987494747989501123056617395, −6.11658136240227932038631602888, −4.83408571729965777185374827687, −4.06962722804624097351802639807, −3.02112894217861435190449591608, −1.73281158551708641052381449247,
0.889396325692088119020345675754, 2.70609444675800472559588073046, 3.98389600702855193801712875269, 5.43741690821658506793918330184, 6.14889978536158819979786214351, 7.20580455426322755028016578321, 7.87965118591515241636831389606, 8.473539114998791054319372722749, 8.741502766749976776856975267265, 10.41802611935113074274986304578