| L(s) = 1 | − 3-s + 4.39·5-s + 7-s − 2·9-s − 2·11-s − 3.39·13-s − 4.39·15-s + 17-s − 6.39·19-s − 21-s + 0.222·23-s + 14.3·25-s + 5·27-s + 2.39·29-s − 6.57·31-s + 2·33-s + 4.39·35-s − 9.97·37-s + 3.39·39-s − 6.39·41-s − 10.5·43-s − 8.79·45-s − 4.22·47-s − 6·49-s − 51-s + 13.3·53-s − 8.79·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.96·5-s + 0.377·7-s − 0.666·9-s − 0.603·11-s − 0.942·13-s − 1.13·15-s + 0.242·17-s − 1.46·19-s − 0.218·21-s + 0.0464·23-s + 2.87·25-s + 0.962·27-s + 0.445·29-s − 1.18·31-s + 0.348·33-s + 0.743·35-s − 1.63·37-s + 0.544·39-s − 0.999·41-s − 1.61·43-s − 1.31·45-s − 0.615·47-s − 0.857·49-s − 0.140·51-s + 1.83·53-s − 1.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 - 4.39T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 19 | \( 1 + 6.39T + 19T^{2} \) |
| 23 | \( 1 - 0.222T + 23T^{2} \) |
| 29 | \( 1 - 2.39T + 29T^{2} \) |
| 31 | \( 1 + 6.57T + 31T^{2} \) |
| 37 | \( 1 + 9.97T + 37T^{2} \) |
| 41 | \( 1 + 6.39T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 4.22T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 61 | \( 1 - 1.39T + 61T^{2} \) |
| 67 | \( 1 + 2.44T + 67T^{2} \) |
| 71 | \( 1 - 6.35T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 + 4.57T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.387536970408179865519597184641, −6.99461622700724687408939768802, −6.59260303077055468452134603944, −5.63128930390334481218375890573, −5.32492384863894211487470495345, −4.68861124239776947928216451303, −3.15291237176619664944276582847, −2.27721176831537020118005293547, −1.64597547128874264796108915571, 0,
1.64597547128874264796108915571, 2.27721176831537020118005293547, 3.15291237176619664944276582847, 4.68861124239776947928216451303, 5.32492384863894211487470495345, 5.63128930390334481218375890573, 6.59260303077055468452134603944, 6.99461622700724687408939768802, 8.387536970408179865519597184641
