| L(s) = 1 | + 2-s + 4-s + 8-s + 4·11-s + 3·13-s + 16-s + 7·17-s + 6·19-s + 4·22-s − 9·23-s + 3·26-s + 3·29-s + 7·31-s + 32-s + 7·34-s − 10·37-s + 6·38-s + 41-s + 13·43-s + 4·44-s − 9·46-s − 2·47-s + 3·52-s + 53-s + 3·58-s + 11·59-s − 13·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.20·11-s + 0.832·13-s + 1/4·16-s + 1.69·17-s + 1.37·19-s + 0.852·22-s − 1.87·23-s + 0.588·26-s + 0.557·29-s + 1.25·31-s + 0.176·32-s + 1.20·34-s − 1.64·37-s + 0.973·38-s + 0.156·41-s + 1.98·43-s + 0.603·44-s − 1.32·46-s − 0.291·47-s + 0.416·52-s + 0.137·53-s + 0.393·58-s + 1.43·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.217350467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.217350467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.73994278121463, −14.83492112176473, −14.14459477264117, −14.05895188656362, −13.65826263828132, −12.72248430572725, −12.08069423142824, −11.92416476647649, −11.47021321450050, −10.43068332949468, −10.23602357416993, −9.448412678089143, −8.931234924277081, −7.995496002442261, −7.765285647306731, −6.908092188612745, −6.273804282549111, −5.822595845287179, −5.240440547868123, −4.402453677935162, −3.687583063311511, −3.416392263470864, −2.466070258220541, −1.449176806587221, −0.9357648134445053,
0.9357648134445053, 1.449176806587221, 2.466070258220541, 3.416392263470864, 3.687583063311511, 4.402453677935162, 5.240440547868123, 5.822595845287179, 6.273804282549111, 6.908092188612745, 7.765285647306731, 7.995496002442261, 8.931234924277081, 9.448412678089143, 10.23602357416993, 10.43068332949468, 11.47021321450050, 11.92416476647649, 12.08069423142824, 12.72248430572725, 13.65826263828132, 14.05895188656362, 14.14459477264117, 14.83492112176473, 15.73994278121463
