| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s − 4·13-s + 16-s + 6·17-s + 4·19-s + 6·22-s + 4·26-s + 6·29-s + 4·31-s − 32-s − 6·34-s − 8·37-s − 4·38-s − 8·43-s − 6·44-s − 4·52-s − 6·53-s − 6·58-s + 6·59-s − 2·61-s − 4·62-s + 64-s + 4·67-s + 6·68-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 1.27·22-s + 0.784·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s − 1.31·37-s − 0.648·38-s − 1.21·43-s − 0.904·44-s − 0.554·52-s − 0.824·53-s − 0.787·58-s + 0.781·59-s − 0.256·61-s − 0.508·62-s + 1/8·64-s + 0.488·67-s + 0.727·68-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
| \(L(1)\) |
\(\approx\) |
\(0.9602604432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9602604432\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−15.57667870916344, −15.24357264337741, −14.38448809615047, −14.05584937607743, −13.34046065856259, −12.67831534222730, −12.20114994517515, −11.76399609308445, −11.05705282705198, −10.26474923746552, −10.04369127853010, −9.736985845160290, −8.711504172907302, −8.242479621543838, −7.636305129474235, −7.352629501835246, −6.583768246162323, −5.745196400667620, −5.095058114164501, −4.858886447803458, −3.521036630545079, −2.938610440225562, −2.401358741436851, −1.413817740995475, −0.4527216493150209,
0.4527216493150209, 1.413817740995475, 2.401358741436851, 2.938610440225562, 3.521036630545079, 4.858886447803458, 5.095058114164501, 5.745196400667620, 6.583768246162323, 7.352629501835246, 7.636305129474235, 8.242479621543838, 8.711504172907302, 9.736985845160290, 10.04369127853010, 10.26474923746552, 11.05705282705198, 11.76399609308445, 12.20114994517515, 12.67831534222730, 13.34046065856259, 14.05584937607743, 14.38448809615047, 15.24357264337741, 15.57667870916344
