| L(s) = 1 | + 1.33·3-s − 2.63·5-s − 1.22·9-s + 3.39·11-s − 2.73·13-s − 3.51·15-s + 17-s + 0.404·19-s + 1.46·23-s + 1.94·25-s − 5.62·27-s + 5.14·29-s + 5.78·31-s + 4.52·33-s + 3.35·37-s − 3.64·39-s − 8.88·41-s − 7.39·43-s + 3.22·45-s − 10.8·47-s + 1.33·51-s − 5.85·53-s − 8.95·55-s + 0.538·57-s + 2.15·59-s − 1.64·61-s + 7.20·65-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 1.17·5-s − 0.407·9-s + 1.02·11-s − 0.758·13-s − 0.906·15-s + 0.242·17-s + 0.0927·19-s + 0.306·23-s + 0.388·25-s − 1.08·27-s + 0.955·29-s + 1.03·31-s + 0.788·33-s + 0.551·37-s − 0.583·39-s − 1.38·41-s − 1.12·43-s + 0.480·45-s − 1.58·47-s + 0.186·51-s − 0.804·53-s − 1.20·55-s + 0.0713·57-s + 0.280·59-s − 0.211·61-s + 0.894·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.33T + 3T^{2} \) |
| 5 | \( 1 + 2.63T + 5T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 19 | \( 1 - 0.404T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 + 8.88T + 41T^{2} \) |
| 43 | \( 1 + 7.39T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 5.85T + 53T^{2} \) |
| 59 | \( 1 - 2.15T + 59T^{2} \) |
| 61 | \( 1 + 1.64T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 5.10T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 1.64T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 5.01T + 89T^{2} \) |
| 97 | \( 1 - 1.08T + 97T^{2} \) |
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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
−8.315186707111921321605841483358, −7.66696051666749108891975936505, −6.91702022948147324302252073559, −6.15302422322882318960843683452, −4.94806417614825631637709724853, −4.28323023889174802219176794391, −3.36260717688569910296952425222, −2.85390847279950945675780157098, −1.50420172457986746489449359819, 0,
1.50420172457986746489449359819, 2.85390847279950945675780157098, 3.36260717688569910296952425222, 4.28323023889174802219176794391, 4.94806417614825631637709724853, 6.15302422322882318960843683452, 6.91702022948147324302252073559, 7.66696051666749108891975936505, 8.315186707111921321605841483358
