| L(s) = 1 | + 5-s − 4·11-s − 13-s + 6·17-s + 4·19-s + 8·23-s + 25-s + 6·29-s + 8·31-s + 10·37-s + 6·41-s + 4·43-s − 7·49-s − 10·53-s − 4·55-s − 4·59-s + 2·61-s − 65-s − 12·67-s + 16·71-s + 2·73-s + 16·79-s + 12·83-s + 6·85-s − 10·89-s + 4·95-s − 6·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 1.64·37-s + 0.937·41-s + 0.609·43-s − 49-s − 1.37·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s − 0.124·65-s − 1.46·67-s + 1.89·71-s + 0.234·73-s + 1.80·79-s + 1.31·83-s + 0.650·85-s − 1.05·89-s + 0.410·95-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.200385042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.200385042\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.83891165962751, −14.25481067764046, −13.89345835192770, −13.27813579696390, −12.78129646176646, −12.35191580803005, −11.77974982793840, −11.03385192634564, −10.70845430960933, −9.986141697419457, −9.630105230799885, −9.182546002938265, −8.258805422611527, −7.812243472253325, −7.506410782581619, −6.590174190644128, −6.144470377631451, −5.376720384279619, −4.986021145404613, −4.477638194095181, −3.329654133973195, −2.897575065693753, −2.419137578866953, −1.205565912806727, −0.7645465119071156,
0.7645465119071156, 1.205565912806727, 2.419137578866953, 2.897575065693753, 3.329654133973195, 4.477638194095181, 4.986021145404613, 5.376720384279619, 6.144470377631451, 6.590174190644128, 7.506410782581619, 7.812243472253325, 8.258805422611527, 9.182546002938265, 9.630105230799885, 9.986141697419457, 10.70845430960933, 11.03385192634564, 11.77974982793840, 12.35191580803005, 12.78129646176646, 13.27813579696390, 13.89345835192770, 14.25481067764046, 14.83891165962751
