| L(s) = 1 | + 27·3-s + 170.·5-s − 803.·7-s + 729·9-s − 8.55e3·11-s − 1.46e3·13-s + 4.59e3·15-s + 3.07e4·17-s − 1.13e4·19-s − 2.16e4·21-s − 1.16e5·23-s − 4.91e4·25-s + 1.96e4·27-s + 1.84e5·29-s + 6.13e4·31-s − 2.31e5·33-s − 1.36e5·35-s + 3.57e5·37-s − 3.95e4·39-s + 3.89e5·41-s + 9.39e5·43-s + 1.24e5·45-s + 1.13e6·47-s − 1.78e5·49-s + 8.31e5·51-s + 3.24e5·53-s − 1.45e6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.609·5-s − 0.885·7-s + 0.333·9-s − 1.93·11-s − 0.185·13-s + 0.351·15-s + 1.51·17-s − 0.381·19-s − 0.511·21-s − 1.99·23-s − 0.628·25-s + 0.192·27-s + 1.40·29-s + 0.369·31-s − 1.11·33-s − 0.539·35-s + 1.15·37-s − 0.106·39-s + 0.882·41-s + 1.80·43-s + 0.203·45-s + 1.60·47-s − 0.216·49-s + 0.877·51-s + 0.299·53-s − 1.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.208059361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208059361\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| good | 5 | \( 1 - 170.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 803.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 8.55e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.46e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.07e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.13e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.16e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.84e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.13e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.57e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.89e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.39e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.13e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.24e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.86e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.14e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.58e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.98e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.61e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.26e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.61e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.25e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.61e6T + 8.07e13T^{2} \) |
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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
−10.00930853215999949513050255213, −9.514738422058690136420743355987, −8.073140827082924723454091337129, −7.68134244031675889040453787055, −6.20031304311768936214002131339, −5.52926914716258696109420377027, −4.15878504538515884473032848267, −2.87919142671959377814331967701, −2.26757520242856909662488006140, −0.64258095646192352351377159147,
0.64258095646192352351377159147, 2.26757520242856909662488006140, 2.87919142671959377814331967701, 4.15878504538515884473032848267, 5.52926914716258696109420377027, 6.20031304311768936214002131339, 7.68134244031675889040453787055, 8.073140827082924723454091337129, 9.514738422058690136420743355987, 10.00930853215999949513050255213
