| L(s) = 1 | − 3·3-s − 7-s + 6·9-s + 2·11-s − 13-s + 2·19-s + 3·21-s − 23-s − 9·27-s + 3·29-s + 31-s − 6·33-s − 2·37-s + 3·39-s − 41-s − 8·43-s − 5·47-s + 49-s − 6·53-s − 6·57-s − 6·61-s − 6·63-s + 10·67-s + 3·69-s − 7·71-s − 13·73-s − 2·77-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s + 0.603·11-s − 0.277·13-s + 0.458·19-s + 0.654·21-s − 0.208·23-s − 1.73·27-s + 0.557·29-s + 0.179·31-s − 1.04·33-s − 0.328·37-s + 0.480·39-s − 0.156·41-s − 1.21·43-s − 0.729·47-s + 1/7·49-s − 0.824·53-s − 0.794·57-s − 0.768·61-s − 0.755·63-s + 1.22·67-s + 0.361·69-s − 0.830·71-s − 1.52·73-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3149024215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3149024215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.72619071788547, −12.13318062270947, −11.97883622937717, −11.42504144736439, −11.17003047844465, −10.50236260355051, −10.12770208412872, −9.794177680304055, −9.271980760006661, −8.663977926667740, −8.085951079396829, −7.449093682028379, −7.003748717074330, −6.526169646593576, −6.238920705282759, −5.689290917149847, −5.196746782152538, −4.703897674529349, −4.343201450081975, −3.583541132169102, −3.134443818995689, −2.270683474611823, −1.466465436804358, −1.103927107625433, −0.1921928758744451,
0.1921928758744451, 1.103927107625433, 1.466465436804358, 2.270683474611823, 3.134443818995689, 3.583541132169102, 4.343201450081975, 4.703897674529349, 5.196746782152538, 5.689290917149847, 6.238920705282759, 6.526169646593576, 7.003748717074330, 7.449093682028379, 8.085951079396829, 8.663977926667740, 9.271980760006661, 9.794177680304055, 10.12770208412872, 10.50236260355051, 11.17003047844465, 11.42504144736439, 11.97883622937717, 12.13318062270947, 12.72619071788547
