| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 4·11-s − 15-s − 2·17-s + 6·19-s − 2·21-s + 25-s − 27-s − 10·29-s − 2·31-s − 4·33-s + 2·35-s − 8·37-s + 2·41-s + 4·43-s + 45-s + 8·47-s − 3·49-s + 2·51-s − 2·53-s + 4·55-s − 6·57-s − 10·61-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.258·15-s − 0.485·17-s + 1.37·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.359·31-s − 0.696·33-s + 0.338·35-s − 1.31·37-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.280·51-s − 0.274·53-s + 0.539·55-s − 0.794·57-s − 1.28·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.628341730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.628341730\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.23168532808396, −12.82718962187507, −12.11708380543491, −11.81297863336528, −11.46895995687169, −10.84393999385013, −10.63014516798379, −9.852124990066370, −9.446749813274228, −8.903393355903769, −8.741928160768772, −7.688308858625476, −7.404755659269546, −7.052218323077981, −6.225532089176976, −5.894936387239209, −5.451810083334677, −4.752805784229592, −4.469559581253055, −3.613130051573669, −3.330773108363609, −2.288685589643816, −1.706197360272170, −1.320123348169866, −0.5029368420567325,
0.5029368420567325, 1.320123348169866, 1.706197360272170, 2.288685589643816, 3.330773108363609, 3.613130051573669, 4.469559581253055, 4.752805784229592, 5.451810083334677, 5.894936387239209, 6.225532089176976, 7.052218323077981, 7.404755659269546, 7.688308858625476, 8.741928160768772, 8.903393355903769, 9.446749813274228, 9.852124990066370, 10.63014516798379, 10.84393999385013, 11.46895995687169, 11.81297863336528, 12.11708380543491, 12.82718962187507, 13.23168532808396
