| L(s) = 1 | − 15.7·2-s − 27·3-s + 118.·4-s + 126.·5-s + 424.·6-s + 1.59e3·7-s + 144.·8-s + 729·9-s − 1.98e3·10-s − 3.50e3·11-s − 3.20e3·12-s − 4.94e3·13-s − 2.50e4·14-s − 3.41e3·15-s − 1.74e4·16-s − 3.23e4·17-s − 1.14e4·18-s − 2.49e4·19-s + 1.50e4·20-s − 4.29e4·21-s + 5.50e4·22-s + 1.78e4·23-s − 3.90e3·24-s − 6.21e4·25-s + 7.76e4·26-s − 1.96e4·27-s + 1.89e5·28-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 0.577·3-s + 0.928·4-s + 0.452·5-s + 0.801·6-s + 1.75·7-s + 0.0999·8-s + 0.333·9-s − 0.628·10-s − 0.794·11-s − 0.535·12-s − 0.624·13-s − 2.43·14-s − 0.261·15-s − 1.06·16-s − 1.59·17-s − 0.462·18-s − 0.835·19-s + 0.419·20-s − 1.01·21-s + 1.10·22-s + 0.305·23-s − 0.0577·24-s − 0.795·25-s + 0.866·26-s − 0.192·27-s + 1.62·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7037138540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7037138540\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 + 15.7T + 128T^{2} \) |
| 5 | \( 1 - 126.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.59e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.50e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.94e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.23e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.78e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.00e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.57e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.02e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.39e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.05e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.89e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.23e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.93e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.51e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.14e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.20e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.71e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.31e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.20e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.45e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.72e7T + 8.07e13T^{2} \) |
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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
−10.09020964614980393482508105614, −8.723120735246785725602699176125, −8.339203989564859558928219795375, −7.36111201709859714320788198531, −6.46553615720876623353289663824, −4.99940670006686373900668027746, −4.59264894223034971656820106262, −2.26824484762266200276811519857, −1.69531170232166199295079290903, −0.47820085797656626673198766776,
0.47820085797656626673198766776, 1.69531170232166199295079290903, 2.26824484762266200276811519857, 4.59264894223034971656820106262, 4.99940670006686373900668027746, 6.46553615720876623353289663824, 7.36111201709859714320788198531, 8.339203989564859558928219795375, 8.723120735246785725602699176125, 10.09020964614980393482508105614
