| L(s) = 1 | + 2-s − 1.64·3-s + 4-s + 2.36·5-s − 1.64·6-s − 2.99·7-s + 8-s − 0.304·9-s + 2.36·10-s + 0.857·11-s − 1.64·12-s + 1.14·13-s − 2.99·14-s − 3.88·15-s + 16-s − 5.38·17-s − 0.304·18-s − 19-s + 2.36·20-s + 4.91·21-s + 0.857·22-s + 1.14·23-s − 1.64·24-s + 0.607·25-s + 1.14·26-s + 5.42·27-s − 2.99·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.947·3-s + 0.5·4-s + 1.05·5-s − 0.670·6-s − 1.13·7-s + 0.353·8-s − 0.101·9-s + 0.748·10-s + 0.258·11-s − 0.473·12-s + 0.316·13-s − 0.799·14-s − 1.00·15-s + 0.250·16-s − 1.30·17-s − 0.0717·18-s − 0.229·19-s + 0.529·20-s + 1.07·21-s + 0.182·22-s + 0.238·23-s − 0.335·24-s + 0.121·25-s + 0.223·26-s + 1.04·27-s − 0.565·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 + 2.99T + 7T^{2} \) |
| 11 | \( 1 - 0.857T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 23 | \( 1 - 1.14T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 + 2.10T + 41T^{2} \) |
| 43 | \( 1 + 8.84T + 43T^{2} \) |
| 47 | \( 1 + 4.62T + 47T^{2} \) |
| 53 | \( 1 - 9.59T + 53T^{2} \) |
| 59 | \( 1 + 6.61T + 59T^{2} \) |
| 61 | \( 1 - 5.14T + 61T^{2} \) |
| 67 | \( 1 + 1.64T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 + 6.51T + 73T^{2} \) |
| 79 | \( 1 - 4.10T + 79T^{2} \) |
| 83 | \( 1 - 2.17T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−6.86899111638728701742491622367, −6.45992591951417262552806820924, −6.30211151881682858204988655528, −5.42200158198642683140016003847, −4.87341775776856988947719621267, −4.03196896766214830557951359368, −3.03490213898255342451470438799, −2.41939433865564649373741795349, −1.29760016692612877674734223292, 0,
1.29760016692612877674734223292, 2.41939433865564649373741795349, 3.03490213898255342451470438799, 4.03196896766214830557951359368, 4.87341775776856988947719621267, 5.42200158198642683140016003847, 6.30211151881682858204988655528, 6.45992591951417262552806820924, 6.86899111638728701742491622367
