L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s + i·9-s + (−1 − i)11-s + 14-s + 16-s − 18-s + (1 − i)22-s − 23-s − i·25-s + i·28-s + (1 − i)29-s + i·32-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s + i·9-s + (−1 − i)11-s + 14-s + 16-s − 18-s + (1 − i)22-s − 23-s − i·25-s + i·28-s + (1 − i)29-s + i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7995561419\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7995561419\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (1 + i)T + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (-1 + i)T - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 + i)T + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (-1 + i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 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
−8.641733391622842512803578234823, −8.037840690588849735231721064313, −7.71152155471619014966592634645, −6.73939598176279258521949409333, −5.98056618051459514259576393545, −5.20151535626506576038568937217, −4.43218783167915691020119632906, −3.62451153843131893607091821528, −2.35340339881131061012398659399, −0.54561443765878412772075480958,
1.43849404969501717806531764121, 2.52640129837559874268583283810, 3.18891518029643972702879888569, 4.28816006298142302012808537098, 5.08090103509392992251434744153, 5.81079809231201425773421817525, 6.78811293983254083951472274207, 7.88301855804167046018481941808, 8.526296817586219861611537572121, 9.319754801670042508729742919002