| L(s) = 1 | + 5-s + 7-s + 13-s + 3·17-s + 19-s + 3·23-s + 25-s − 3·29-s − 2·31-s + 35-s + 10·37-s − 6·41-s + 2·43-s − 6·49-s + 3·53-s − 3·59-s − 8·61-s + 65-s − 7·67-s + 12·71-s − 13·73-s − 14·79-s − 6·83-s + 3·85-s − 6·89-s + 91-s + 95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.277·13-s + 0.727·17-s + 0.229·19-s + 0.625·23-s + 1/5·25-s − 0.557·29-s − 0.359·31-s + 0.169·35-s + 1.64·37-s − 0.937·41-s + 0.304·43-s − 6/7·49-s + 0.412·53-s − 0.390·59-s − 1.02·61-s + 0.124·65-s − 0.855·67-s + 1.42·71-s − 1.52·73-s − 1.57·79-s − 0.658·83-s + 0.325·85-s − 0.635·89-s + 0.104·91-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.73005086048048, −14.17721160202023, −13.66421700826821, −13.11602061532459, −12.77500642479460, −12.09837321774314, −11.56695739185529, −11.10947699767783, −10.60123475162238, −10.00690669598835, −9.495922347979111, −9.069918060531283, −8.382785135703500, −7.914916730891531, −7.316625956126544, −6.820800737491836, −6.011998896196311, −5.708865021399140, −5.029277525330287, −4.474021709137856, −3.774533032361012, −3.057884605989019, −2.531211308155768, −1.556656691612980, −1.179394299638481, 0,
1.179394299638481, 1.556656691612980, 2.531211308155768, 3.057884605989019, 3.774533032361012, 4.474021709137856, 5.029277525330287, 5.708865021399140, 6.011998896196311, 6.820800737491836, 7.316625956126544, 7.914916730891531, 8.382785135703500, 9.069918060531283, 9.495922347979111, 10.00690669598835, 10.60123475162238, 11.10947699767783, 11.56695739185529, 12.09837321774314, 12.77500642479460, 13.11602061532459, 13.66421700826821, 14.17721160202023, 14.73005086048048
