| L(s) = 1 | − 2·7-s − 3·11-s − 3·13-s + 2·17-s + 4·19-s + 5·23-s + 6·31-s + 3·37-s + 2·41-s − 4·43-s − 5·47-s − 3·49-s − 4·53-s + 9·59-s + 3·61-s + 2·67-s + 9·71-s − 14·73-s + 6·77-s − 14·79-s − 14·89-s + 6·91-s − 7·97-s − 10·101-s − 14·103-s − 107-s − 10·109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.904·11-s − 0.832·13-s + 0.485·17-s + 0.917·19-s + 1.04·23-s + 1.07·31-s + 0.493·37-s + 0.312·41-s − 0.609·43-s − 0.729·47-s − 3/7·49-s − 0.549·53-s + 1.17·59-s + 0.384·61-s + 0.244·67-s + 1.06·71-s − 1.63·73-s + 0.683·77-s − 1.57·79-s − 1.48·89-s + 0.628·91-s − 0.710·97-s − 0.995·101-s − 1.37·103-s − 0.0966·107-s − 0.957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.81829608724101127382809331595, −7.07925097869552427728918944331, −6.50957969876926047939462948106, −5.48370077040940809203551126787, −5.07522290082727202194661952881, −4.10159679424389859934466788749, −2.99620435373602248089097840184, −2.70150780702509304849832522189, −1.25214759116015842743487209919, 0,
1.25214759116015842743487209919, 2.70150780702509304849832522189, 2.99620435373602248089097840184, 4.10159679424389859934466788749, 5.07522290082727202194661952881, 5.48370077040940809203551126787, 6.50957969876926047939462948106, 7.07925097869552427728918944331, 7.81829608724101127382809331595
