| L(s) = 1 | + 2-s + 4-s + 8-s + 6·11-s + 2·13-s + 16-s + 2·17-s − 4·19-s + 6·22-s − 4·23-s + 2·26-s + 2·29-s + 2·31-s + 32-s + 2·34-s + 10·37-s − 4·38-s − 6·41-s + 2·43-s + 6·44-s − 4·46-s − 2·47-s + 2·52-s + 6·53-s + 2·58-s + 4·59-s + 12·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 1.27·22-s − 0.834·23-s + 0.392·26-s + 0.371·29-s + 0.359·31-s + 0.176·32-s + 0.342·34-s + 1.64·37-s − 0.648·38-s − 0.937·41-s + 0.304·43-s + 0.904·44-s − 0.589·46-s − 0.291·47-s + 0.277·52-s + 0.824·53-s + 0.262·58-s + 0.520·59-s + 1.53·61-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
| \(L(1)\) |
\(\approx\) |
\(4.671150067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.671150067\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.41768010190307, −14.82048588586638, −14.45103887353133, −14.01959498444108, −13.36689507088014, −12.89196523238762, −12.19214694850918, −11.79241938699531, −11.35632499198758, −10.72845620301169, −9.951818723550478, −9.607761278074425, −8.642806748334233, −8.429821687429969, −7.530896533965452, −6.855379767103865, −6.279624937555949, −5.991711980043198, −5.128145277636657, −4.212246627625215, −4.034624513258857, −3.283863895628806, −2.388166435707766, −1.596560529912340, −0.8271623825473100,
0.8271623825473100, 1.596560529912340, 2.388166435707766, 3.283863895628806, 4.034624513258857, 4.212246627625215, 5.128145277636657, 5.991711980043198, 6.279624937555949, 6.855379767103865, 7.530896533965452, 8.429821687429969, 8.642806748334233, 9.607761278074425, 9.951818723550478, 10.72845620301169, 11.35632499198758, 11.79241938699531, 12.19214694850918, 12.89196523238762, 13.36689507088014, 14.01959498444108, 14.45103887353133, 14.82048588586638, 15.41768010190307
