L(s) = 1 | − 3-s − 7-s + 9-s − 3·11-s + 3·13-s − 5·19-s + 21-s − 4·23-s − 27-s − 5·29-s + 2·31-s + 3·33-s + 10·37-s − 3·39-s − 43-s − 3·47-s − 6·49-s − 4·53-s + 5·57-s − 4·59-s − 8·61-s − 63-s + 2·67-s + 4·69-s + 16·73-s + 3·77-s + 14·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.832·13-s − 1.14·19-s + 0.218·21-s − 0.834·23-s − 0.192·27-s − 0.928·29-s + 0.359·31-s + 0.522·33-s + 1.64·37-s − 0.480·39-s − 0.152·43-s − 0.437·47-s − 6/7·49-s − 0.549·53-s + 0.662·57-s − 0.520·59-s − 1.02·61-s − 0.125·63-s + 0.244·67-s + 0.481·69-s + 1.87·73-s + 0.341·77-s + 1.57·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.82475801164550, −14.20007777833540, −13.57615490327924, −13.12010224257208, −12.73994088361279, −12.31426033573791, −11.52374835668820, −11.12278418950134, −10.72496849944517, −10.12806317663809, −9.665401374193673, −9.080035283988180, −8.374028374481052, −7.894267666359807, −7.459116481197565, −6.545267410336715, −6.174714494834507, −5.858779985910989, −4.946530375576886, −4.577909773067957, −3.776684733677443, −3.275121749970777, −2.350524511931686, −1.816005790218706, −0.7780072610956329, 0,
0.7780072610956329, 1.816005790218706, 2.350524511931686, 3.275121749970777, 3.776684733677443, 4.577909773067957, 4.946530375576886, 5.858779985910989, 6.174714494834507, 6.545267410336715, 7.459116481197565, 7.894267666359807, 8.374028374481052, 9.080035283988180, 9.665401374193673, 10.12806317663809, 10.72496849944517, 11.12278418950134, 11.52374835668820, 12.31426033573791, 12.73994088361279, 13.12010224257208, 13.57615490327924, 14.20007777833540, 14.82475801164550