| L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s − 6·13-s + 16-s − 2·17-s − 4·22-s − 6·26-s − 6·29-s − 8·31-s + 32-s − 2·34-s + 10·37-s + 2·41-s − 4·43-s − 4·44-s − 8·47-s − 6·52-s − 2·53-s − 6·58-s − 8·59-s + 14·61-s − 8·62-s + 64-s + 12·67-s − 2·68-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.852·22-s − 1.17·26-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s + 1.64·37-s + 0.312·41-s − 0.609·43-s − 0.603·44-s − 1.16·47-s − 0.832·52-s − 0.274·53-s − 0.787·58-s − 1.04·59-s + 1.79·61-s − 1.01·62-s + 1/8·64-s + 1.46·67-s − 0.242·68-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
| \(L(1)\) |
\(\approx\) |
\(1.842974556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.842974556\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.38716780641268, −14.97037922487492, −14.44324528167688, −14.04492975802030, −13.12105133167618, −12.76457433077132, −12.68884377529301, −11.56976781543707, −11.34836723138370, −10.71343058538970, −9.944378908161039, −9.659649053989728, −8.891804338028523, −7.941304274751658, −7.678926491648713, −7.044336333131375, −6.419710590317103, −5.558200675576844, −5.162785822598946, −4.636085217564663, −3.853396567982809, −3.089607628972951, −2.352025437728617, −1.936781269115221, −0.4532593565291576,
0.4532593565291576, 1.936781269115221, 2.352025437728617, 3.089607628972951, 3.853396567982809, 4.636085217564663, 5.162785822598946, 5.558200675576844, 6.419710590317103, 7.044336333131375, 7.678926491648713, 7.941304274751658, 8.891804338028523, 9.659649053989728, 9.944378908161039, 10.71343058538970, 11.34836723138370, 11.56976781543707, 12.68884377529301, 12.76457433077132, 13.12105133167618, 14.04492975802030, 14.44324528167688, 14.97037922487492, 15.38716780641268
