| L(s) = 1 | − 81·3-s − 1.35e3·5-s − 4.62e3·7-s + 6.56e3·9-s − 3.55e4·11-s − 4.84e4·13-s + 1.09e5·15-s − 1.30e5·17-s + 5.61e5·19-s + 3.74e5·21-s − 4.00e5·23-s − 1.29e5·25-s − 5.31e5·27-s + 3.25e6·29-s − 3.79e5·31-s + 2.87e6·33-s + 6.25e6·35-s + 1.76e7·37-s + 3.92e6·39-s + 4.50e6·41-s − 2.91e7·43-s − 8.86e6·45-s + 3.36e7·47-s − 1.89e7·49-s + 1.05e7·51-s + 5.68e6·53-s + 4.80e7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.966·5-s − 0.728·7-s + 0.333·9-s − 0.732·11-s − 0.470·13-s + 0.557·15-s − 0.378·17-s + 0.988·19-s + 0.420·21-s − 0.298·23-s − 0.0661·25-s − 0.192·27-s + 0.853·29-s − 0.0738·31-s + 0.422·33-s + 0.704·35-s + 1.55·37-s + 0.271·39-s + 0.249·41-s − 1.29·43-s − 0.322·45-s + 1.00·47-s − 0.469·49-s + 0.218·51-s + 0.0988·53-s + 0.707·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 81T \) |
| good | 5 | \( 1 + 1.35e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.62e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.55e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 4.84e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.30e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.61e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 4.00e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.25e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.79e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.76e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.50e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.91e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.36e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.68e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 8.79e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.07e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.95e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.89e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.62e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.10e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.60e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.44e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.59e8T + 7.60e17T^{2} \) |
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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
−9.558749219518581040906173654888, −8.262381918701033093955092609815, −7.47595496971153986014496236709, −6.60482062508142432657615452707, −5.51980552153942977745741726276, −4.54329600427176312110668771429, −3.52776463430214895631583154263, −2.46234406386209354725024368350, −0.823063173700572891650597203094, 0,
0.823063173700572891650597203094, 2.46234406386209354725024368350, 3.52776463430214895631583154263, 4.54329600427176312110668771429, 5.51980552153942977745741726276, 6.60482062508142432657615452707, 7.47595496971153986014496236709, 8.262381918701033093955092609815, 9.558749219518581040906173654888
