| L(s) = 1 | − 3-s − 3·7-s + 9-s + 4·11-s − 3·13-s − 5·19-s + 3·21-s − 6·23-s − 27-s − 29-s − 5·31-s − 4·33-s + 6·37-s + 3·39-s − 5·41-s − 43-s + 6·47-s + 2·49-s + 6·53-s + 5·57-s + 14·59-s − 61-s − 3·63-s − 11·67-s + 6·69-s + 11·73-s − 12·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.20·11-s − 0.832·13-s − 1.14·19-s + 0.654·21-s − 1.25·23-s − 0.192·27-s − 0.185·29-s − 0.898·31-s − 0.696·33-s + 0.986·37-s + 0.480·39-s − 0.780·41-s − 0.152·43-s + 0.875·47-s + 2/7·49-s + 0.824·53-s + 0.662·57-s + 1.82·59-s − 0.128·61-s − 0.377·63-s − 1.34·67-s + 0.722·69-s + 1.28·73-s − 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.19167764395148, −12.67296141169063, −12.33624313966325, −11.94161671797112, −11.46243635103525, −10.99841761844069, −10.19646173284820, −10.16537671744019, −9.552769575113280, −9.077831287319130, −8.713103088468580, −7.945245640424147, −7.480189908766168, −6.808675025937148, −6.583709321193990, −6.106266326650906, −5.624188351645903, −5.032768263432363, −4.317498388098315, −3.885477295217707, −3.573817180115711, −2.624134148336125, −2.179025935322633, −1.479893311516027, −0.6064094543400013, 0,
0.6064094543400013, 1.479893311516027, 2.179025935322633, 2.624134148336125, 3.573817180115711, 3.885477295217707, 4.317498388098315, 5.032768263432363, 5.624188351645903, 6.106266326650906, 6.583709321193990, 6.808675025937148, 7.480189908766168, 7.945245640424147, 8.713103088468580, 9.077831287319130, 9.552769575113280, 10.16537671744019, 10.19646173284820, 10.99841761844069, 11.46243635103525, 11.94161671797112, 12.33624313966325, 12.67296141169063, 13.19167764395148
