| 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
| \(L(1)\) |
\(\approx\) |
\(3.204612459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.204612459\) |
| \(L(\frac{3}{2})\) |
|
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
−15.67737203942957, −14.70148191500476, −14.47503301720127, −14.04771267755115, −13.37756554019278, −12.66697001804696, −12.45691017784454, −11.82460621314126, −11.25659727443670, −10.66003454677444, −9.949896674581976, −9.696914169189745, −8.857596652139183, −7.989030098841215, −7.600183566970490, −7.176699986842220, −6.165935293602439, −5.813523564885220, −5.128001259656688, −4.559947343327426, −3.851322587372997, −3.087965752777853, −2.521763778363656, −1.708495165265559, −0.6366007001170947,
0.6366007001170947, 1.708495165265559, 2.521763778363656, 3.087965752777853, 3.851322587372997, 4.559947343327426, 5.128001259656688, 5.813523564885220, 6.165935293602439, 7.176699986842220, 7.600183566970490, 7.989030098841215, 8.857596652139183, 9.696914169189745, 9.949896674581976, 10.66003454677444, 11.25659727443670, 11.82460621314126, 12.45691017784454, 12.66697001804696, 13.37756554019278, 14.04771267755115, 14.47503301720127, 14.70148191500476, 15.67737203942957
