| L(s) = 1 | − 2·3-s + 2·5-s − 7-s + 9-s + 2·13-s − 4·15-s − 2·17-s − 4·19-s + 2·21-s − 25-s + 4·27-s + 2·29-s + 10·31-s − 2·35-s − 8·37-s − 4·39-s + 2·41-s − 4·43-s + 2·45-s + 6·47-s + 49-s + 4·51-s + 12·53-s + 8·57-s + 10·59-s + 6·61-s − 63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.03·15-s − 0.485·17-s − 0.917·19-s + 0.436·21-s − 1/5·25-s + 0.769·27-s + 0.371·29-s + 1.79·31-s − 0.338·35-s − 1.31·37-s − 0.640·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s + 0.875·47-s + 1/7·49-s + 0.560·51-s + 1.64·53-s + 1.05·57-s + 1.30·59-s + 0.768·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.31671381820014, −12.54058150300708, −12.19708236577316, −11.67109800404728, −11.42315671855939, −10.61394452333568, −10.42494952135043, −10.13316159986352, −9.492840777488622, −8.895981795335713, −8.513117550803300, −8.113491791429640, −7.169267902416785, −6.724862921824559, −6.472566647890210, −5.904330987482616, −5.560630397657538, −5.140901509760719, −4.342212247920921, −4.121262407489313, −3.232061757508022, −2.594051986731048, −2.121428956821000, −1.343305309803312, −0.7373708693721311, 0,
0.7373708693721311, 1.343305309803312, 2.121428956821000, 2.594051986731048, 3.232061757508022, 4.121262407489313, 4.342212247920921, 5.140901509760719, 5.560630397657538, 5.904330987482616, 6.472566647890210, 6.724862921824559, 7.169267902416785, 8.113491791429640, 8.513117550803300, 8.895981795335713, 9.492840777488622, 10.13316159986352, 10.42494952135043, 10.61394452333568, 11.42315671855939, 11.67109800404728, 12.19708236577316, 12.54058150300708, 13.31671381820014
