L(s) = 1 | − 2·4-s + 5-s − 2·7-s + 2·11-s + 13-s + 4·16-s + 2·19-s − 2·20-s + 6·23-s + 25-s + 4·28-s − 5·29-s + 3·31-s − 2·35-s + 4·37-s + 10·41-s + 10·43-s − 4·44-s + 12·47-s − 3·49-s − 2·52-s − 3·53-s + 2·55-s − 59-s − 2·61-s − 8·64-s + 65-s + ⋯ |
L(s) = 1 | − 4-s + 0.447·5-s − 0.755·7-s + 0.603·11-s + 0.277·13-s + 16-s + 0.458·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s + 0.755·28-s − 0.928·29-s + 0.538·31-s − 0.338·35-s + 0.657·37-s + 1.56·41-s + 1.52·43-s − 0.603·44-s + 1.75·47-s − 3/7·49-s − 0.277·52-s − 0.412·53-s + 0.269·55-s − 0.130·59-s − 0.256·61-s − 64-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.399901562\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.399901562\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.22237772465965, −12.86315968540447, −12.43049649452751, −11.94595315870280, −11.19009841838257, −10.90698046751523, −10.22336364971822, −9.814649007846925, −9.250787307682476, −9.085366348037162, −8.693788330371024, −7.903852675213992, −7.329972052162586, −7.065824213066291, −6.047485702560741, −5.964952867163093, −5.494265271484784, −4.535401879811033, −4.416839424313525, −3.721042466850806, −3.048349967593548, −2.735901915405286, −1.722354080023216, −1.016744199897080, −0.5524969385363875,
0.5524969385363875, 1.016744199897080, 1.722354080023216, 2.735901915405286, 3.048349967593548, 3.721042466850806, 4.416839424313525, 4.535401879811033, 5.494265271484784, 5.964952867163093, 6.047485702560741, 7.065824213066291, 7.329972052162586, 7.903852675213992, 8.693788330371024, 9.085366348037162, 9.250787307682476, 9.814649007846925, 10.22336364971822, 10.90698046751523, 11.19009841838257, 11.94595315870280, 12.43049649452751, 12.86315968540447, 13.22237772465965