| L(s) = 1 | − 2·3-s + 5-s + 2·7-s + 9-s + 2·13-s − 2·15-s + 6·17-s + 4·19-s − 4·21-s − 6·23-s + 25-s + 4·27-s + 6·29-s + 4·31-s + 2·35-s − 2·37-s − 4·39-s − 6·41-s + 10·43-s + 45-s + 6·47-s − 3·49-s − 12·51-s + 6·53-s − 8·57-s + 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.338·35-s − 0.328·37-s − 0.640·39-s − 0.937·41-s + 1.52·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s − 1.68·51-s + 0.824·53-s − 1.05·57-s + 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.408635214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.408635214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.66564426892317, −14.16817894166901, −13.95557143309774, −13.32006430375039, −12.46496000128118, −12.07820471789166, −11.79726157974608, −11.18696361440652, −10.66474028649155, −10.08836670748914, −9.826873101641211, −8.959777040846896, −8.216132037143170, −8.003779725885989, −7.108720507124592, −6.609416213180197, −5.836698407845197, −5.598942070444495, −5.089648718867403, −4.400174385159265, −3.677469085138930, −2.890682516290387, −2.060882857383472, −1.133462339701602, −0.7413270921589541,
0.7413270921589541, 1.133462339701602, 2.060882857383472, 2.890682516290387, 3.677469085138930, 4.400174385159265, 5.089648718867403, 5.598942070444495, 5.836698407845197, 6.609416213180197, 7.108720507124592, 8.003779725885989, 8.216132037143170, 8.959777040846896, 9.826873101641211, 10.08836670748914, 10.66474028649155, 11.18696361440652, 11.79726157974608, 12.07820471789166, 12.46496000128118, 13.32006430375039, 13.95557143309774, 14.16817894166901, 14.66564426892317
