| L(s) = 1 | + 22.1·2-s + 363.·4-s − 258.·5-s + 411.·7-s + 5.21e3·8-s − 5.73e3·10-s − 437.·11-s + 1.32e4·13-s + 9.11e3·14-s + 6.91e4·16-s − 2.49e4·17-s + 3.80e4·19-s − 9.40e4·20-s − 9.70e3·22-s − 7.97e4·23-s − 1.11e4·25-s + 2.93e5·26-s + 1.49e5·28-s + 6.43e4·29-s + 2.41e5·31-s + 8.65e5·32-s − 5.52e5·34-s − 1.06e5·35-s + 3.53e5·37-s + 8.42e5·38-s − 1.35e6·40-s − 2.46e5·41-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 2.83·4-s − 0.925·5-s + 0.452·7-s + 3.60·8-s − 1.81·10-s − 0.0991·11-s + 1.66·13-s + 0.887·14-s + 4.22·16-s − 1.23·17-s + 1.27·19-s − 2.62·20-s − 0.194·22-s − 1.36·23-s − 0.142·25-s + 3.27·26-s + 1.28·28-s + 0.489·29-s + 1.45·31-s + 4.67·32-s − 2.41·34-s − 0.419·35-s + 1.14·37-s + 2.49·38-s − 3.33·40-s − 0.557·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(9.568113032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.568113032\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 22.1T + 128T^{2} \) |
| 5 | \( 1 + 258.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 411.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 437.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.32e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.49e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.80e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.97e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.43e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.41e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.53e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.46e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.51e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.41e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 8.46e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.36e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.07e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.42e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.19e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.96e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.48e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.97e6T + 8.07e13T^{2} \) |
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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
−10.15225261512916378535791392609, −8.356208732265486497075184111766, −7.71104351736262280562361675998, −6.60624829097380366718654166597, −5.93754929647779490702771665663, −4.81933166075912604452059170939, −4.07681309565795847311156697204, −3.40624527137058513958494967641, −2.25073213825132885897561550772, −1.05689306341803478848423363729,
1.05689306341803478848423363729, 2.25073213825132885897561550772, 3.40624527137058513958494967641, 4.07681309565795847311156697204, 4.81933166075912604452059170939, 5.93754929647779490702771665663, 6.60624829097380366718654166597, 7.71104351736262280562361675998, 8.356208732265486497075184111766, 10.15225261512916378535791392609
