| L(s) = 1 | + 135.·3-s − 581. i·5-s − 2.67e3i·7-s + 1.19e4·9-s + 1.41e4·11-s − 4.28e4i·13-s − 7.90e4i·15-s − 1.40e5·17-s − 1.29e5·19-s − 3.62e5i·21-s − 1.96e5i·23-s + 5.22e4·25-s + 7.27e5·27-s + 3.55e5i·29-s − 3.35e3i·31-s + ⋯ |
| L(s) = 1 | + 1.67·3-s − 0.930i·5-s − 1.11i·7-s + 1.81·9-s + 0.966·11-s − 1.49i·13-s − 1.56i·15-s − 1.68·17-s − 0.995·19-s − 1.86i·21-s − 0.703i·23-s + 0.133·25-s + 1.36·27-s + 0.503i·29-s − 0.00363i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.626473177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.626473177\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 135.T + 6.56e3T^{2} \) |
| 5 | \( 1 + 581. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 2.67e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.41e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 4.28e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 1.40e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.29e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.96e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 3.55e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 3.35e3iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 9.07e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 3.06e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 5.01e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 3.25e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 2.34e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 7.78e6T + 1.46e14T^{2} \) |
| 61 | \( 1 - 2.42e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 6.87e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 5.76e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.19e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.55e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.03e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 1.19e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 3.19e7T + 7.83e15T^{2} \) |
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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
−10.04636945259129298142750222508, −8.907480754306974132310878640684, −8.546318294854243963749997685772, −7.55181701644956071103990773617, −6.51901732166014102930311494038, −4.63900800620921156775106095369, −3.98753153430122294268816230408, −2.84092474403497556980037671009, −1.59201979098791445211898500333, −0.54805441984793063227691174220,
1.95261851293614295795973677925, 2.30113276891567705639189903108, 3.53090349473259072774034364990, 4.42788076286553235401792752984, 6.40331276876496387764974249082, 6.97003758477321810831310855690, 8.322981362794337154140564143820, 9.063192755833561062806203217355, 9.499062189842931621593931019468, 10.96216923558218515664339517435
