| L(s) = 1 | + 0.656·2-s + 1.91·3-s − 1.56·4-s − 3.56·5-s + 1.25·6-s − 2.34·8-s + 0.656·9-s − 2.34·10-s − 11-s − 3·12-s − 5.91·13-s − 6.82·15-s + 1.59·16-s − 1.65·17-s + 0.431·18-s + 1.48·19-s + 5.59·20-s − 0.656·22-s + 3.34·23-s − 4.48·24-s + 7.73·25-s − 3.88·26-s − 4.48·27-s + 3.08·29-s − 4.48·30-s − 7.08·31-s + 5.73·32-s + ⋯ |
| L(s) = 1 | + 0.464·2-s + 1.10·3-s − 0.784·4-s − 1.59·5-s + 0.512·6-s − 0.828·8-s + 0.218·9-s − 0.741·10-s − 0.301·11-s − 0.866·12-s − 1.63·13-s − 1.76·15-s + 0.399·16-s − 0.401·17-s + 0.101·18-s + 0.339·19-s + 1.25·20-s − 0.139·22-s + 0.697·23-s − 0.914·24-s + 1.54·25-s − 0.761·26-s − 0.862·27-s + 0.571·29-s − 0.818·30-s − 1.27·31-s + 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.656T + 2T^{2} \) |
| 3 | \( 1 - 1.91T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 - 3.34T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 + 7.08T + 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 - 1.59T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 - 9.22T + 53T^{2} \) |
| 59 | \( 1 - 8.85T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 + 9.82T + 67T^{2} \) |
| 71 | \( 1 + 8.61T + 71T^{2} \) |
| 73 | \( 1 - 4.56T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 - 0.167T + 83T^{2} \) |
| 89 | \( 1 + 2.56T + 89T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
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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
−10.26658329915847082761187630619, −9.190855776298553568879598828402, −8.622332400215452707349614317324, −7.74828989340668861098029346259, −7.14115808936953489511206754834, −5.31710467468675038827123040827, −4.45016504790278300255613480830, −3.55903607239581580550562159406, −2.71504829467129263430489083099, 0,
2.71504829467129263430489083099, 3.55903607239581580550562159406, 4.45016504790278300255613480830, 5.31710467468675038827123040827, 7.14115808936953489511206754834, 7.74828989340668861098029346259, 8.622332400215452707349614317324, 9.190855776298553568879598828402, 10.26658329915847082761187630619
