| L(s) = 1 | + 3-s + 3·7-s + 9-s + 11-s + 13-s − 3·17-s + 4·19-s + 3·21-s + 2·23-s + 27-s − 3·29-s + 5·31-s + 33-s + 2·37-s + 39-s + 2·43-s + 9·47-s + 2·49-s − 3·51-s + 5·53-s + 4·57-s + 7·59-s + 11·61-s + 3·63-s + 3·67-s + 2·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.727·17-s + 0.917·19-s + 0.654·21-s + 0.417·23-s + 0.192·27-s − 0.557·29-s + 0.898·31-s + 0.174·33-s + 0.328·37-s + 0.160·39-s + 0.304·43-s + 1.31·47-s + 2/7·49-s − 0.420·51-s + 0.686·53-s + 0.529·57-s + 0.911·59-s + 1.40·61-s + 0.377·63-s + 0.366·67-s + 0.240·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.648590543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.648590543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.21422972431463, −13.87376960307369, −13.32559618185380, −12.92324643485983, −12.13031974484133, −11.74852933061946, −11.20924438181859, −10.82235848287532, −10.17656004502235, −9.565926508116514, −9.064778640497701, −8.627061794121630, −7.994413360096802, −7.731095240219481, −6.910714572952645, −6.637133466466701, −5.631191398801003, −5.284692594914974, −4.557285733076008, −4.050986634310438, −3.496061364396601, −2.596223713376685, −2.180808515990784, −1.324940061514065, −0.7675079292766684,
0.7675079292766684, 1.324940061514065, 2.180808515990784, 2.596223713376685, 3.496061364396601, 4.050986634310438, 4.557285733076008, 5.284692594914974, 5.631191398801003, 6.637133466466701, 6.910714572952645, 7.731095240219481, 7.994413360096802, 8.627061794121630, 9.064778640497701, 9.565926508116514, 10.17656004502235, 10.82235848287532, 11.20924438181859, 11.74852933061946, 12.13031974484133, 12.92324643485983, 13.32559618185380, 13.87376960307369, 14.21422972431463
