L(s) = 1 | + 3-s + 3·7-s − 2·9-s + 3·11-s + 2·17-s + 2·19-s + 3·21-s + 6·23-s − 5·25-s − 5·27-s − 2·29-s + 4·31-s + 3·33-s + 37-s + 7·41-s − 4·43-s − 47-s + 2·49-s + 2·51-s + 9·53-s + 2·57-s − 8·59-s − 4·61-s − 6·63-s − 12·67-s + 6·69-s + 5·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.904·11-s + 0.485·17-s + 0.458·19-s + 0.654·21-s + 1.25·23-s − 25-s − 0.962·27-s − 0.371·29-s + 0.718·31-s + 0.522·33-s + 0.164·37-s + 1.09·41-s − 0.609·43-s − 0.145·47-s + 2/7·49-s + 0.280·51-s + 1.23·53-s + 0.264·57-s − 1.04·59-s − 0.512·61-s − 0.755·63-s − 1.46·67-s + 0.722·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.987193761\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.987193761\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.82405209250554057086839096190, −9.596579125586026945584894800190, −8.904993800674481517136818658058, −8.080337610010359288480922886283, −7.36102788145509220927936510098, −6.07828897597428129385301535102, −5.10085747284785110782395778247, −3.97768080475343207584093772636, −2.81288670918314267095866007614, −1.43453783211503200534897488154,
1.43453783211503200534897488154, 2.81288670918314267095866007614, 3.97768080475343207584093772636, 5.10085747284785110782395778247, 6.07828897597428129385301535102, 7.36102788145509220927936510098, 8.080337610010359288480922886283, 8.904993800674481517136818658058, 9.596579125586026945584894800190, 10.82405209250554057086839096190