| L(s) = 1 | + 3·5-s + 4·7-s − 11-s − 17-s + 4·19-s − 3·23-s + 4·25-s − 3·29-s + 12·35-s − 2·37-s + 2·41-s − 43-s − 4·47-s + 9·49-s − 3·55-s + 2·59-s − 8·61-s − 5·67-s − 4·77-s − 8·79-s − 3·85-s + 6·89-s + 12·95-s − 12·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.51·7-s − 0.301·11-s − 0.242·17-s + 0.917·19-s − 0.625·23-s + 4/5·25-s − 0.557·29-s + 2.02·35-s − 0.328·37-s + 0.312·41-s − 0.152·43-s − 0.583·47-s + 9/7·49-s − 0.404·55-s + 0.260·59-s − 1.02·61-s − 0.610·67-s − 0.455·77-s − 0.900·79-s − 0.325·85-s + 0.635·89-s + 1.23·95-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.83319128484446, −13.67378613238427, −13.14977116868020, −12.48252602699379, −12.01438684738302, −11.39558831695254, −11.11306039965104, −10.42480636154247, −10.14453927961702, −9.429533144119777, −9.181526893113834, −8.432508944414825, −8.044371832707103, −7.491930214030319, −6.985194427871059, −6.233336907462283, −5.763257161885355, −5.293717916477636, −4.890121580918220, −4.281738165652890, −3.559717142301885, −2.700668059010489, −2.237404026528188, −1.522578248291845, −1.275326291188472, 0,
1.275326291188472, 1.522578248291845, 2.237404026528188, 2.700668059010489, 3.559717142301885, 4.281738165652890, 4.890121580918220, 5.293717916477636, 5.763257161885355, 6.233336907462283, 6.985194427871059, 7.491930214030319, 8.044371832707103, 8.432508944414825, 9.181526893113834, 9.429533144119777, 10.14453927961702, 10.42480636154247, 11.11306039965104, 11.39558831695254, 12.01438684738302, 12.48252602699379, 13.14977116868020, 13.67378613238427, 13.83319128484446
