| L(s) = 1 | + 3·7-s − 3·9-s − 3·11-s − 17-s − 4·19-s + 2·23-s + 7·29-s − 5·31-s − 6·37-s − 4·41-s + 6·43-s + 13·47-s + 2·49-s + 9·53-s − 5·59-s + 13·61-s − 9·63-s + 5·67-s + 2·73-s − 9·77-s − 14·79-s + 9·81-s + 9·83-s − 4·89-s + 2·97-s + 9·99-s + 101-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 9-s − 0.904·11-s − 0.242·17-s − 0.917·19-s + 0.417·23-s + 1.29·29-s − 0.898·31-s − 0.986·37-s − 0.624·41-s + 0.914·43-s + 1.89·47-s + 2/7·49-s + 1.23·53-s − 0.650·59-s + 1.66·61-s − 1.13·63-s + 0.610·67-s + 0.234·73-s − 1.02·77-s − 1.57·79-s + 81-s + 0.987·83-s − 0.423·89-s + 0.203·97-s + 0.904·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.07757480241250, −15.57316262865702, −15.02400789795008, −14.47880365569838, −14.04828896028082, −13.51445562028070, −12.81423754184250, −12.25347861550505, −11.64401542075760, −11.11241287133122, −10.58408555669516, −10.24982605617720, −9.155446686594639, −8.640894031543728, −8.318115725459285, −7.622873833681176, −7.008314505872074, −6.228787926915841, −5.430927864746791, −5.138912772857764, −4.354771334160163, −3.603348624486852, −2.565023536916380, −2.240628756641519, −1.098859956801154, 0,
1.098859956801154, 2.240628756641519, 2.565023536916380, 3.603348624486852, 4.354771334160163, 5.138912772857764, 5.430927864746791, 6.228787926915841, 7.008314505872074, 7.622873833681176, 8.318115725459285, 8.640894031543728, 9.155446686594639, 10.24982605617720, 10.58408555669516, 11.11241287133122, 11.64401542075760, 12.25347861550505, 12.81423754184250, 13.51445562028070, 14.04828896028082, 14.47880365569838, 15.02400789795008, 15.57316262865702, 16.07757480241250
