| L(s) = 1 | + 200.·5-s − 288.·7-s + 3.48e3·11-s − 54.9·13-s − 1.96e4·17-s − 3.44e4·19-s + 6.00e4·23-s − 3.80e4·25-s − 3.86e3·29-s − 7.77e4·31-s − 5.77e4·35-s + 5.22e5·37-s + 2.01e5·41-s − 9.43e5·43-s − 5.45e5·47-s − 7.40e5·49-s − 1.50e6·53-s + 6.96e5·55-s + 2.63e6·59-s − 1.95e5·61-s − 1.09e4·65-s + 2.70e6·67-s − 4.32e6·71-s + 3.99e6·73-s − 1.00e6·77-s − 5.41e5·79-s − 1.58e6·83-s + ⋯ |
| L(s) = 1 | + 0.716·5-s − 0.317·7-s + 0.788·11-s − 0.00693·13-s − 0.968·17-s − 1.15·19-s + 1.02·23-s − 0.487·25-s − 0.0293·29-s − 0.468·31-s − 0.227·35-s + 1.69·37-s + 0.455·41-s − 1.80·43-s − 0.766·47-s − 0.898·49-s − 1.39·53-s + 0.564·55-s + 1.67·59-s − 0.110·61-s − 0.00496·65-s + 1.09·67-s − 1.43·71-s + 1.20·73-s − 0.250·77-s − 0.123·79-s − 0.304·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 200.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 288.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.48e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 54.9T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.96e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.44e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.00e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.86e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.77e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.22e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.01e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.45e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.50e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.63e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.95e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.70e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.32e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.99e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.41e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.58e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.32e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.21e5T + 8.07e13T^{2} \) |
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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
−9.786363435688275501632696119758, −9.161124203914723382166971634711, −8.165198299589185717417756670754, −6.74984369705195112179888943227, −6.24442294910559161982892858620, −4.96060053074827594980097285122, −3.85557298385712610338127657162, −2.51251385238350712738129439661, −1.45801592957720685027873348361, 0,
1.45801592957720685027873348361, 2.51251385238350712738129439661, 3.85557298385712610338127657162, 4.96060053074827594980097285122, 6.24442294910559161982892858620, 6.74984369705195112179888943227, 8.165198299589185717417756670754, 9.161124203914723382166971634711, 9.786363435688275501632696119758
