| L(s) = 1 | − 7-s + 4·11-s − 2·13-s + 2·17-s + 4·19-s + 8·23-s − 2·29-s + 6·37-s + 6·41-s + 4·43-s + 49-s + 10·53-s − 12·59-s − 14·61-s + 12·67-s − 8·71-s − 10·73-s − 4·77-s − 16·79-s − 12·83-s − 10·89-s + 2·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s − 0.371·29-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 1.37·53-s − 1.56·59-s − 1.79·61-s + 1.46·67-s − 0.949·71-s − 1.17·73-s − 0.455·77-s − 1.80·79-s − 1.31·83-s − 1.05·89-s + 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.03372765435064, −13.55731229982980, −12.86568546650484, −12.61829051655925, −12.03849471237292, −11.49215688801503, −11.21878266930634, −10.54271882486923, −9.930504155935126, −9.543028049976538, −8.999928724864621, −8.807180353359876, −7.849735586019741, −7.351382509599863, −7.100131654118174, −6.388016209745256, −5.834039556700819, −5.419869808698137, −4.597220954944976, −4.248395238568076, −3.487460117842548, −2.951750509046548, −2.471623712051385, −1.301402534592914, −1.123638906223401, 0,
1.123638906223401, 1.301402534592914, 2.471623712051385, 2.951750509046548, 3.487460117842548, 4.248395238568076, 4.597220954944976, 5.419869808698137, 5.834039556700819, 6.388016209745256, 7.100131654118174, 7.351382509599863, 7.849735586019741, 8.807180353359876, 8.999928724864621, 9.543028049976538, 9.930504155935126, 10.54271882486923, 11.21878266930634, 11.49215688801503, 12.03849471237292, 12.61829051655925, 12.86568546650484, 13.55731229982980, 14.03372765435064
