| L(s) = 1 | − 1.73·2-s + 0.999·4-s + 3.46·5-s − 4·7-s + 1.73·8-s − 5.99·10-s − 5.19·11-s − 4·13-s + 6.92·14-s − 5·16-s + 3.46·17-s + 19-s + 3.46·20-s + 9·22-s − 3.46·23-s + 6.99·25-s + 6.92·26-s − 3.99·28-s + 1.73·29-s − 7·31-s + 5.19·32-s − 5.99·34-s − 13.8·35-s − 10·37-s − 1.73·38-s + 6.00·40-s − 1.73·41-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s + 1.54·5-s − 1.51·7-s + 0.612·8-s − 1.89·10-s − 1.56·11-s − 1.10·13-s + 1.85·14-s − 1.25·16-s + 0.840·17-s + 0.229·19-s + 0.774·20-s + 1.91·22-s − 0.722·23-s + 1.39·25-s + 1.35·26-s − 0.755·28-s + 0.321·29-s − 1.25·31-s + 0.918·32-s − 1.02·34-s − 2.34·35-s − 1.64·37-s − 0.280·38-s + 0.948·40-s − 0.270·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 + 8.66T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.10318943519972514752225404301, −9.720586260790132553080187634736, −9.018466007880534133555257610799, −7.80742178424810234766040540342, −6.99492906534982146295916815255, −5.88795701931733664889609899124, −5.05964804985766465605336763313, −3.04664608694473858439904046720, −1.97403135362735832229470712239, 0,
1.97403135362735832229470712239, 3.04664608694473858439904046720, 5.05964804985766465605336763313, 5.88795701931733664889609899124, 6.99492906534982146295916815255, 7.80742178424810234766040540342, 9.018466007880534133555257610799, 9.720586260790132553080187634736, 10.10318943519972514752225404301
