| L(s) = 1 | + 2-s + 4-s − 5-s + 3·7-s + 8-s − 10-s − 3·11-s − 2·13-s + 3·14-s + 16-s − 8·17-s − 7·19-s − 20-s − 3·22-s − 7·23-s + 25-s − 2·26-s + 3·28-s + 8·29-s − 31-s + 32-s − 8·34-s − 3·35-s − 4·37-s − 7·38-s − 40-s + 43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s − 0.554·13-s + 0.801·14-s + 1/4·16-s − 1.94·17-s − 1.60·19-s − 0.223·20-s − 0.639·22-s − 1.45·23-s + 1/5·25-s − 0.392·26-s + 0.566·28-s + 1.48·29-s − 0.179·31-s + 0.176·32-s − 1.37·34-s − 0.507·35-s − 0.657·37-s − 1.13·38-s − 0.158·40-s + 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.206799610505433868450018584390, −7.81193722214417873281060328553, −6.74409318801140412787407651448, −6.19284717019496646388198927064, −4.87596174050881724410164289287, −4.71716814632695587336211253831, −3.83173526298716506960710350833, −2.48221200077289387244327586334, −1.94654126829090639479835146422, 0,
1.94654126829090639479835146422, 2.48221200077289387244327586334, 3.83173526298716506960710350833, 4.71716814632695587336211253831, 4.87596174050881724410164289287, 6.19284717019496646388198927064, 6.74409318801140412787407651448, 7.81193722214417873281060328553, 8.206799610505433868450018584390
