| L(s) = 1 | + 3·5-s + 2·7-s − 5·13-s + 3·17-s + 2·19-s + 6·23-s + 4·25-s − 3·29-s − 2·31-s + 6·35-s − 7·37-s + 3·41-s + 8·43-s + 6·47-s − 3·49-s + 3·53-s + 10·61-s − 15·65-s + 10·67-s + 12·71-s − 14·73-s + 2·79-s − 18·83-s + 9·85-s + 9·89-s − 10·91-s + 6·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.755·7-s − 1.38·13-s + 0.727·17-s + 0.458·19-s + 1.25·23-s + 4/5·25-s − 0.557·29-s − 0.359·31-s + 1.01·35-s − 1.15·37-s + 0.468·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.412·53-s + 1.28·61-s − 1.86·65-s + 1.22·67-s + 1.42·71-s − 1.63·73-s + 0.225·79-s − 1.97·83-s + 0.976·85-s + 0.953·89-s − 1.04·91-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.384179382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.384179382\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−15.83656444878677, −15.16765514819072, −14.53574754829337, −14.25685463017594, −13.84837221652211, −12.91479888290161, −12.76439624417254, −11.95336650170553, −11.40084077629022, −10.71965393406950, −10.17949550635690, −9.638215995738775, −9.190688923082308, −8.575145117615151, −7.680551182265922, −7.273653817597623, −6.643866238066823, −5.676940301043994, −5.357016808326862, −4.883748401530168, −3.963966264280398, −2.982428687412027, −2.326155904233769, −1.685157108318777, −0.7979257003612947,
0.7979257003612947, 1.685157108318777, 2.326155904233769, 2.982428687412027, 3.963966264280398, 4.883748401530168, 5.357016808326862, 5.676940301043994, 6.643866238066823, 7.273653817597623, 7.680551182265922, 8.575145117615151, 9.190688923082308, 9.638215995738775, 10.17949550635690, 10.71965393406950, 11.40084077629022, 11.95336650170553, 12.76439624417254, 12.91479888290161, 13.84837221652211, 14.25685463017594, 14.53574754829337, 15.16765514819072, 15.83656444878677
