| L(s) = 1 | − 38.1·2-s + 135.·3-s + 944.·4-s − 33.6·5-s − 5.16e3·6-s + 4.81e3·7-s − 1.65e4·8-s − 1.37e3·9-s + 1.28e3·10-s + 5.44e3·11-s + 1.27e5·12-s − 8.45e4·13-s − 1.83e5·14-s − 4.55e3·15-s + 1.46e5·16-s + 5.25e4·18-s + 7.31e5·19-s − 3.18e4·20-s + 6.51e5·21-s − 2.07e5·22-s − 1.21e6·23-s − 2.23e6·24-s − 1.95e6·25-s + 3.22e6·26-s − 2.84e6·27-s + 4.55e6·28-s + 4.60e6·29-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.964·3-s + 1.84·4-s − 0.0241·5-s − 1.62·6-s + 0.758·7-s − 1.42·8-s − 0.0699·9-s + 0.0406·10-s + 0.112·11-s + 1.77·12-s − 0.821·13-s − 1.27·14-s − 0.0232·15-s + 0.560·16-s + 0.117·18-s + 1.28·19-s − 0.0444·20-s + 0.731·21-s − 0.189·22-s − 0.903·23-s − 1.37·24-s − 0.999·25-s + 1.38·26-s − 1.03·27-s + 1.39·28-s + 1.20·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 + 38.1T + 512T^{2} \) |
| 3 | \( 1 - 135.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 33.6T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.81e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.44e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 8.45e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 7.31e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.21e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.60e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.07e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.10e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.46e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.54e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.04e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.14e5T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.32e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.38e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 4.10e6T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.18e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.79e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.27e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.00e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.12e9T + 7.60e17T^{2} \) |
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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
−9.670514819631135334544299264736, −8.744852516172948734997947988994, −7.949543855479758277742349825778, −7.58728203308357409561700910075, −6.23810609562100778911549094926, −4.69298050194672812929767424314, −3.07561670600260323866003986150, −2.16860287639591083504475567123, −1.22940387293496942618314364016, 0,
1.22940387293496942618314364016, 2.16860287639591083504475567123, 3.07561670600260323866003986150, 4.69298050194672812929767424314, 6.23810609562100778911549094926, 7.58728203308357409561700910075, 7.949543855479758277742349825778, 8.744852516172948734997947988994, 9.670514819631135334544299264736
