| L(s) = 1 | − 3-s + 2·5-s + 9-s − 13-s − 2·15-s − 17-s + 4·23-s − 25-s − 27-s − 6·29-s + 4·31-s − 8·37-s + 39-s + 10·43-s + 2·45-s + 8·47-s − 7·49-s + 51-s + 6·53-s − 12·59-s + 12·61-s − 2·65-s + 6·67-s − 4·69-s + 14·73-s + 75-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.277·13-s − 0.516·15-s − 0.242·17-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 1.31·37-s + 0.160·39-s + 1.52·43-s + 0.298·45-s + 1.16·47-s − 49-s + 0.140·51-s + 0.824·53-s − 1.56·59-s + 1.53·61-s − 0.248·65-s + 0.733·67-s − 0.481·69-s + 1.63·73-s + 0.115·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.79430587975967, −12.35108253800161, −12.07775666310462, −11.42243421372574, −10.86645104514251, −10.75326031281568, −10.13280541636895, −9.638161264935266, −9.201544990235653, −9.016755132716933, −8.092300988023596, −7.850942389184672, −7.135708996198313, −6.679194191138780, −6.422077166461182, −5.602502932033179, −5.453706176871779, −5.004010048094508, −4.279329113114363, −3.846927790055289, −3.196108059393634, −2.407988266741773, −2.141734378723537, −1.382425427325754, −0.8070682625520731, 0,
0.8070682625520731, 1.382425427325754, 2.141734378723537, 2.407988266741773, 3.196108059393634, 3.846927790055289, 4.279329113114363, 5.004010048094508, 5.453706176871779, 5.602502932033179, 6.422077166461182, 6.679194191138780, 7.135708996198313, 7.850942389184672, 8.092300988023596, 9.016755132716933, 9.201544990235653, 9.638161264935266, 10.13280541636895, 10.75326031281568, 10.86645104514251, 11.42243421372574, 12.07775666310462, 12.35108253800161, 12.79430587975967
