| L(s) = 1 | + 2-s − 18·3-s − 31·4-s − 16·5-s − 18·6-s + 28·7-s − 63·8-s + 81·9-s − 16·10-s − 138·11-s + 558·12-s + 82·13-s + 28·14-s + 288·15-s + 929·16-s − 289·17-s + 81·18-s − 2.26e3·19-s + 496·20-s − 504·21-s − 138·22-s − 3.42e3·23-s + 1.13e3·24-s − 2.86e3·25-s + 82·26-s + 2.91e3·27-s − 868·28-s + ⋯ |
| L(s) = 1 | + 0.176·2-s − 1.15·3-s − 0.968·4-s − 0.286·5-s − 0.204·6-s + 0.215·7-s − 0.348·8-s + 1/3·9-s − 0.0505·10-s − 0.343·11-s + 1.11·12-s + 0.134·13-s + 0.0381·14-s + 0.330·15-s + 0.907·16-s − 0.242·17-s + 0.0589·18-s − 1.43·19-s + 0.277·20-s − 0.249·21-s − 0.0607·22-s − 1.34·23-s + 0.401·24-s − 0.918·25-s + 0.0237·26-s + 0.769·27-s − 0.209·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - T + p^{5} T^{2} \) |
| 3 | \( 1 + 2 p^{2} T + p^{5} T^{2} \) |
| 5 | \( 1 + 16 T + p^{5} T^{2} \) |
| 7 | \( 1 - 4 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 138 T + p^{5} T^{2} \) |
| 13 | \( 1 - 82 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2260 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3424 T + p^{5} T^{2} \) |
| 29 | \( 1 - 8304 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4580 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5932 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9990 T + p^{5} T^{2} \) |
| 43 | \( 1 + 12776 T + p^{5} T^{2} \) |
| 47 | \( 1 + 768 T + p^{5} T^{2} \) |
| 53 | \( 1 + 12630 T + p^{5} T^{2} \) |
| 59 | \( 1 - 37968 T + p^{5} T^{2} \) |
| 61 | \( 1 - 18476 T + p^{5} T^{2} \) |
| 67 | \( 1 + 51272 T + p^{5} T^{2} \) |
| 71 | \( 1 + 10592 T + p^{5} T^{2} \) |
| 73 | \( 1 + 70974 T + p^{5} T^{2} \) |
| 79 | \( 1 + 25944 T + p^{5} T^{2} \) |
| 83 | \( 1 + 63056 T + p^{5} T^{2} \) |
| 89 | \( 1 - 7706 T + p^{5} T^{2} \) |
| 97 | \( 1 - 99662 T + p^{5} T^{2} \) |
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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
−17.43489817728147599481556862973, −16.07923624576371922303899640796, −14.45564965424066469198243901407, −12.95414466834622342614704991977, −11.72171435633495141964463498431, −10.27042021003922031951078185534, −8.351889877357219617174733933439, −6.01866163907878141228557939370, −4.47029489904721083727467527060, 0,
4.47029489904721083727467527060, 6.01866163907878141228557939370, 8.351889877357219617174733933439, 10.27042021003922031951078185534, 11.72171435633495141964463498431, 12.95414466834622342614704991977, 14.45564965424066469198243901407, 16.07923624576371922303899640796, 17.43489817728147599481556862973
