| L(s) = 1 | − 5-s − 2·11-s + 3·13-s + 17-s + 2·19-s + 4·23-s − 4·25-s − 3·29-s + 31-s − 4·37-s − 3·41-s + 2·43-s − 47-s − 6·53-s + 2·55-s − 11·59-s − 2·61-s − 3·65-s + 4·67-s + 10·71-s + 2·73-s + 16·79-s + 3·83-s − 85-s − 2·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s + 0.832·13-s + 0.242·17-s + 0.458·19-s + 0.834·23-s − 4/5·25-s − 0.557·29-s + 0.179·31-s − 0.657·37-s − 0.468·41-s + 0.304·43-s − 0.145·47-s − 0.824·53-s + 0.269·55-s − 1.43·59-s − 0.256·61-s − 0.372·65-s + 0.488·67-s + 1.18·71-s + 0.234·73-s + 1.80·79-s + 0.329·83-s − 0.108·85-s − 0.205·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.74089959813885, −13.42396469298490, −12.79113440446213, −12.38758630349385, −11.88783599749098, −11.31869234264717, −10.92780057944799, −10.56453316363828, −9.862357025974813, −9.419239283444778, −8.917548232616131, −8.330236853649469, −7.811015274068497, −7.559820301186684, −6.813513062241591, −6.342165947673749, −5.744970280246068, −5.165639471131433, −4.768935307711408, −3.964306809736865, −3.489285528059849, −3.053363030390378, −2.235935645409269, −1.568361059983523, −0.8256166502174795, 0,
0.8256166502174795, 1.568361059983523, 2.235935645409269, 3.053363030390378, 3.489285528059849, 3.964306809736865, 4.768935307711408, 5.165639471131433, 5.744970280246068, 6.342165947673749, 6.813513062241591, 7.559820301186684, 7.811015274068497, 8.330236853649469, 8.917548232616131, 9.419239283444778, 9.862357025974813, 10.56453316363828, 10.92780057944799, 11.31869234264717, 11.88783599749098, 12.38758630349385, 12.79113440446213, 13.42396469298490, 13.74089959813885
