| L(s) = 1 | + 2.21e11·3-s − 3.08e16·5-s − 1.11e20·7-s + 2.26e22·9-s + 1.19e24·11-s − 2.71e24·13-s − 6.84e27·15-s + 1.17e29·17-s + 1.65e30·19-s − 2.46e31·21-s − 5.41e31·23-s + 2.40e32·25-s − 8.72e32·27-s − 1.03e34·29-s + 3.80e34·31-s + 2.65e35·33-s + 3.42e36·35-s + 8.41e36·37-s − 6.02e35·39-s + 7.68e37·41-s + 2.17e38·43-s − 6.98e38·45-s − 2.73e39·47-s + 7.09e39·49-s + 2.60e40·51-s + 8.29e39·53-s − 3.68e40·55-s + ⋯ |
| L(s) = 1 | + 1.36·3-s − 1.15·5-s − 1.53·7-s + 0.852·9-s + 0.402·11-s − 0.0180·13-s − 1.57·15-s + 1.42·17-s + 1.47·19-s − 2.08·21-s − 0.541·23-s + 0.338·25-s − 0.201·27-s − 0.445·29-s + 0.341·31-s + 0.547·33-s + 1.77·35-s + 1.18·37-s − 0.0245·39-s + 0.966·41-s + 0.891·43-s − 0.985·45-s − 1.39·47-s + 1.35·49-s + 1.94·51-s + 0.250·53-s − 0.465·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+47/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(24)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{49}{2})\) |
|
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 | \( 1 - 2.21e11T + 2.65e22T^{2} \) |
| 5 | \( 1 + 3.08e16T + 7.10e32T^{2} \) |
| 7 | \( 1 + 1.11e20T + 5.24e39T^{2} \) |
| 11 | \( 1 - 1.19e24T + 8.81e48T^{2} \) |
| 13 | \( 1 + 2.71e24T + 2.26e52T^{2} \) |
| 17 | \( 1 - 1.17e29T + 6.77e57T^{2} \) |
| 19 | \( 1 - 1.65e30T + 1.26e60T^{2} \) |
| 23 | \( 1 + 5.41e31T + 1.00e64T^{2} \) |
| 29 | \( 1 + 1.03e34T + 5.40e68T^{2} \) |
| 31 | \( 1 - 3.80e34T + 1.24e70T^{2} \) |
| 37 | \( 1 - 8.41e36T + 5.07e73T^{2} \) |
| 41 | \( 1 - 7.68e37T + 6.32e75T^{2} \) |
| 43 | \( 1 - 2.17e38T + 5.92e76T^{2} \) |
| 47 | \( 1 + 2.73e39T + 3.87e78T^{2} \) |
| 53 | \( 1 - 8.29e39T + 1.09e81T^{2} \) |
| 59 | \( 1 + 7.09e40T + 1.69e83T^{2} \) |
| 61 | \( 1 + 6.43e41T + 8.13e83T^{2} \) |
| 67 | \( 1 + 1.19e43T + 6.69e85T^{2} \) |
| 71 | \( 1 + 2.64e43T + 1.02e87T^{2} \) |
| 73 | \( 1 - 3.87e43T + 3.76e87T^{2} \) |
| 79 | \( 1 - 1.22e44T + 1.54e89T^{2} \) |
| 83 | \( 1 + 1.65e45T + 1.57e90T^{2} \) |
| 89 | \( 1 + 6.00e45T + 4.18e91T^{2} \) |
| 97 | \( 1 + 1.75e46T + 2.38e93T^{2} \) |
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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
−9.883683663766925843170804034384, −9.256922163587197727034280424443, −7.947698563235009285742006197398, −7.35548906665726781824642605964, −5.91088649146271843248510197561, −4.08120357435325617258550795575, −3.34630890604671075809091529738, −2.81000190041243396312116947255, −1.15733683538810549237334056439, 0,
1.15733683538810549237334056439, 2.81000190041243396312116947255, 3.34630890604671075809091529738, 4.08120357435325617258550795575, 5.91088649146271843248510197561, 7.35548906665726781824642605964, 7.947698563235009285742006197398, 9.256922163587197727034280424443, 9.883683663766925843170804034384
