| L(s) = 1 | + 27·3-s + 230.·5-s + 729·9-s − 5.02e3·11-s + 1.37e4·13-s + 6.22e3·15-s − 3.24e4·17-s − 9.65e3·19-s + 3.20e4·23-s − 2.50e4·25-s + 1.96e4·27-s + 1.03e5·29-s − 2.41e5·31-s − 1.35e5·33-s − 1.27e5·37-s + 3.71e5·39-s − 6.07e5·41-s + 4.43e5·43-s + 1.67e5·45-s − 6.91e5·47-s − 8.76e5·51-s + 6.72e5·53-s − 1.15e6·55-s − 2.60e5·57-s + 2.58e6·59-s − 1.53e6·61-s + 3.17e6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.824·5-s + 0.333·9-s − 1.13·11-s + 1.73·13-s + 0.475·15-s − 1.60·17-s − 0.322·19-s + 0.548·23-s − 0.320·25-s + 0.192·27-s + 0.789·29-s − 1.45·31-s − 0.657·33-s − 0.412·37-s + 1.00·39-s − 1.37·41-s + 0.851·43-s + 0.274·45-s − 0.971·47-s − 0.925·51-s + 0.620·53-s − 0.938·55-s − 0.186·57-s + 1.63·59-s − 0.866·61-s + 1.43·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 27T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 230.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 5.02e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.37e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.24e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.65e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.20e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.03e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.41e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.27e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.07e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.91e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.72e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.58e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.53e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.20e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.51e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.47e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.75e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.06e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.798662546833700358922201995876, −8.646411207577870492610944207758, −7.35972993514765062641935221386, −6.39626558153849916078332449177, −5.56302387563977546706794249061, −4.44262303808077935854672547105, −3.32705302034246304111447564702, −2.29047745092843433565209089328, −1.47197259788083612472288540204, 0,
1.47197259788083612472288540204, 2.29047745092843433565209089328, 3.32705302034246304111447564702, 4.44262303808077935854672547105, 5.56302387563977546706794249061, 6.39626558153849916078332449177, 7.35972993514765062641935221386, 8.646411207577870492610944207758, 8.798662546833700358922201995876
