| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s − 18·5-s − 8·6-s − 2·7-s + 8·8-s − 11·9-s − 36·10-s − 52·11-s − 16·12-s + 28·13-s − 4·14-s + 72·15-s + 16·16-s + 14·17-s − 22·18-s − 16·19-s − 72·20-s + 8·21-s − 104·22-s − 36·23-s − 32·24-s + 199·25-s + 56·26-s + 152·27-s − 8·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 1/2·4-s − 1.60·5-s − 0.544·6-s − 0.107·7-s + 0.353·8-s − 0.407·9-s − 1.13·10-s − 1.42·11-s − 0.384·12-s + 0.597·13-s − 0.0763·14-s + 1.23·15-s + 1/4·16-s + 0.199·17-s − 0.288·18-s − 0.193·19-s − 0.804·20-s + 0.0831·21-s − 1.00·22-s − 0.326·23-s − 0.272·24-s + 1.59·25-s + 0.422·26-s + 1.08·27-s − 0.0539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 41 | \( 1 + p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 16 T + p^{3} T^{2} \) |
| 23 | \( 1 + 36 T + p^{3} T^{2} \) |
| 29 | \( 1 + 160 T + p^{3} T^{2} \) |
| 31 | \( 1 - 132 T + p^{3} T^{2} \) |
| 37 | \( 1 + 294 T + p^{3} T^{2} \) |
| 43 | \( 1 - 356 T + p^{3} T^{2} \) |
| 47 | \( 1 - 42 T + p^{3} T^{2} \) |
| 53 | \( 1 + 548 T + p^{3} T^{2} \) |
| 59 | \( 1 - 252 T + p^{3} T^{2} \) |
| 61 | \( 1 + 494 T + p^{3} T^{2} \) |
| 67 | \( 1 + 616 T + p^{3} T^{2} \) |
| 71 | \( 1 - 738 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1010 T + p^{3} T^{2} \) |
| 79 | \( 1 + 834 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1436 T + p^{3} T^{2} \) |
| 89 | \( 1 - 474 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1598 T + p^{3} T^{2} \) |
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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
−13.05781731404672694988677726001, −12.14089579751385523204551781364, −11.30542295452777821091554262735, −10.56655197214834909933591659818, −8.383523366578419877750545151696, −7.39436361471170781681857266305, −5.89329925347244055635800467268, −4.65964345912732488363115619409, −3.22804614496871000854302290278, 0,
3.22804614496871000854302290278, 4.65964345912732488363115619409, 5.89329925347244055635800467268, 7.39436361471170781681857266305, 8.383523366578419877750545151696, 10.56655197214834909933591659818, 11.30542295452777821091554262735, 12.14089579751385523204551781364, 13.05781731404672694988677726001
