| L(s) = 1 | + 3.35·2-s + 27·3-s − 116.·4-s + 0.849·5-s + 90.4·6-s + 405.·7-s − 820.·8-s + 729·9-s + 2.84·10-s + 2.66e3·11-s − 3.15e3·12-s − 9.50e3·13-s + 1.35e3·14-s + 22.9·15-s + 1.21e4·16-s − 3.61e4·17-s + 2.44e3·18-s − 3.58e4·19-s − 99.1·20-s + 1.09e4·21-s + 8.92e3·22-s − 1.21e4·23-s − 2.21e4·24-s − 7.81e4·25-s − 3.18e4·26-s + 1.96e4·27-s − 4.73e4·28-s + ⋯ |
| L(s) = 1 | + 0.296·2-s + 0.577·3-s − 0.912·4-s + 0.00303·5-s + 0.171·6-s + 0.447·7-s − 0.566·8-s + 0.333·9-s + 0.000899·10-s + 0.603·11-s − 0.526·12-s − 1.20·13-s + 0.132·14-s + 0.00175·15-s + 0.744·16-s − 1.78·17-s + 0.0987·18-s − 1.19·19-s − 0.00277·20-s + 0.258·21-s + 0.178·22-s − 0.208·23-s − 0.327·24-s − 0.999·25-s − 0.355·26-s + 0.192·27-s − 0.407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 3.35T + 128T^{2} \) |
| 5 | \( 1 - 0.849T + 7.81e4T^{2} \) |
| 7 | \( 1 - 405.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.66e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.50e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.61e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.58e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.28e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.26e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.83e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 7.56e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.69e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.47e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.25e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.26e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.62e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.11e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.91e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.45e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.88e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.34e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.52e7T + 8.07e13T^{2} \) |
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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
−12.93382497889186957636814229626, −11.85698559347632378488250933444, −10.28184341055251390584020800570, −9.101598037843497637562214000524, −8.291144267567985855638366146232, −6.71436215027444129279417425193, −4.92776973769270564030534028978, −3.95000786673034120206117976458, −2.13329574007374215183964643362, 0,
2.13329574007374215183964643362, 3.95000786673034120206117976458, 4.92776973769270564030534028978, 6.71436215027444129279417425193, 8.291144267567985855638366146232, 9.101598037843497637562214000524, 10.28184341055251390584020800570, 11.85698559347632378488250933444, 12.93382497889186957636814229626
