L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 5-s − 0.999·8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)16-s + 1.73i·17-s + 19-s + (−0.499 + 0.866i)20-s − 23-s + 25-s − 1.73i·31-s + (0.499 − 0.866i)32-s + (−1.49 + 0.866i)34-s + (0.5 + 0.866i)38-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 5-s − 0.999·8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)16-s + 1.73i·17-s + 19-s + (−0.499 + 0.866i)20-s − 23-s + 25-s − 1.73i·31-s + (0.499 − 0.866i)32-s + (−1.49 + 0.866i)34-s + (0.5 + 0.866i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.434449014\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.434449014\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - 1.73iT - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.73iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 2T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 1.73iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.73iT - T^{2} \) |
| 83 | \( 1 + 1.73iT - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−9.967094511630002132362033973609, −9.501704922783443986962260404163, −8.397507380657265063768272422563, −7.84204171831906188317733810386, −6.67597027772975196912052189636, −6.03987122400830339625076356985, −5.38717823918188904608958474212, −4.30448062350206923087347667575, −3.29848818436494529445248891710, −1.93598480651231218226739935785,
1.34225732997095697318256003625, 2.54832594156346273710871363919, 3.37979763031487280795803323210, 4.80442440001438324094990063770, 5.31259196135914591160031334551, 6.28957581810395958815907928466, 7.19244977660992586847552213955, 8.542128382364495245659577516935, 9.416313602098014119626519050808, 9.886282222828552205782785037272