| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s + 4·11-s + 2·13-s + 2·15-s − 6·17-s − 4·19-s + 21-s − 23-s − 25-s + 27-s + 2·29-s − 8·31-s + 4·33-s + 2·35-s − 6·37-s + 2·39-s − 6·41-s + 4·43-s + 2·45-s − 8·47-s + 49-s − 6·51-s − 6·53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s − 0.917·19-s + 0.218·21-s − 0.208·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.338·35-s − 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.19942093159856, −14.75650853400172, −14.27926641368770, −13.84600192837762, −13.24765435570506, −12.98952079571088, −12.20638978688179, −11.60522342944015, −11.05174132441280, −10.53258632923169, −9.952004817943698, −9.231621618985570, −8.925866546574860, −8.520071459489707, −7.786957293185510, −6.994644563085540, −6.483970292759363, −6.122375049325342, −5.271486820400391, −4.587300844243689, −3.959674969308091, −3.431933376402704, −2.413992834565775, −1.827125722142128, −1.422095868063140, 0,
1.422095868063140, 1.827125722142128, 2.413992834565775, 3.431933376402704, 3.959674969308091, 4.587300844243689, 5.271486820400391, 6.122375049325342, 6.483970292759363, 6.994644563085540, 7.786957293185510, 8.520071459489707, 8.925866546574860, 9.231621618985570, 9.952004817943698, 10.53258632923169, 11.05174132441280, 11.60522342944015, 12.20638978688179, 12.98952079571088, 13.24765435570506, 13.84600192837762, 14.27926641368770, 14.75650853400172, 15.19942093159856
