| L(s) = 1 | − 2-s + 4-s − 8-s − 2·11-s + 3·13-s + 16-s − 4·17-s + 3·19-s + 2·22-s + 6·23-s − 3·26-s + 6·29-s − 8·31-s − 32-s + 4·34-s − 5·37-s − 3·38-s + 10·41-s − 4·43-s − 2·44-s − 6·46-s + 2·47-s + 3·52-s + 8·53-s − 6·58-s + 2·59-s − 61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.603·11-s + 0.832·13-s + 1/4·16-s − 0.970·17-s + 0.688·19-s + 0.426·22-s + 1.25·23-s − 0.588·26-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.685·34-s − 0.821·37-s − 0.486·38-s + 1.56·41-s − 0.609·43-s − 0.301·44-s − 0.884·46-s + 0.291·47-s + 0.416·52-s + 1.09·53-s − 0.787·58-s + 0.260·59-s − 0.128·61-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
| \(L(1)\) |
\(\approx\) |
\(1.433146260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.433146260\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−15.65689309189949, −15.13838196930849, −14.55613456886232, −13.89634087633793, −13.18111887845009, −13.01455418312310, −12.12872201955234, −11.61458122325697, −10.96492748762107, −10.63765250787444, −10.10479402613444, −9.200830363005254, −8.937620440255764, −8.420767199309207, −7.599062406189923, −7.207718779554462, −6.553454742221736, −5.881286660904311, −5.243932129181298, −4.549324623736274, −3.669228169458164, −2.987530038085377, −2.282991448629857, −1.397666743599006, −0.5704805167730888,
0.5704805167730888, 1.397666743599006, 2.282991448629857, 2.987530038085377, 3.669228169458164, 4.549324623736274, 5.243932129181298, 5.881286660904311, 6.553454742221736, 7.207718779554462, 7.599062406189923, 8.420767199309207, 8.937620440255764, 9.200830363005254, 10.10479402613444, 10.63765250787444, 10.96492748762107, 11.61458122325697, 12.12872201955234, 13.01455418312310, 13.18111887845009, 13.89634087633793, 14.55613456886232, 15.13838196930849, 15.65689309189949
