| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s + 407.·5-s − 216·6-s − 513.·7-s − 512·8-s + 729·9-s − 3.26e3·10-s + 1.81e3·11-s + 1.72e3·12-s − 2.64e3·13-s + 4.10e3·14-s + 1.10e4·15-s + 4.09e3·16-s − 2.56e4·17-s − 5.83e3·18-s + 8.06e3·19-s + 2.60e4·20-s − 1.38e4·21-s − 1.45e4·22-s − 3.23e4·23-s − 1.38e4·24-s + 8.80e4·25-s + 2.11e4·26-s + 1.96e4·27-s − 3.28e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.45·5-s − 0.408·6-s − 0.565·7-s − 0.353·8-s + 0.333·9-s − 1.03·10-s + 0.411·11-s + 0.288·12-s − 0.334·13-s + 0.400·14-s + 0.841·15-s + 0.250·16-s − 1.26·17-s − 0.235·18-s + 0.269·19-s + 0.729·20-s − 0.326·21-s − 0.291·22-s − 0.555·23-s − 0.204·24-s + 1.12·25-s + 0.236·26-s + 0.192·27-s − 0.282·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 5 | \( 1 - 407.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 513.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.81e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.64e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.56e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 8.06e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.23e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.19e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.49e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 9.48e3T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.43e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.64e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.17e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.23e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 5.14e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.37e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.75e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.45e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.58e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.11e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.63e6T + 8.07e13T^{2} \) |
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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
−9.607451970572798908743601803746, −9.168928442540544782523503572020, −8.157821683798138923936886508683, −6.84064873815925500996451684206, −6.29225256592074273092921297952, −5.02108718104169309201523950529, −3.42852830524443739199824395804, −2.26193663834795188260337642259, −1.57520673320971122526197718853, 0,
1.57520673320971122526197718853, 2.26193663834795188260337642259, 3.42852830524443739199824395804, 5.02108718104169309201523950529, 6.29225256592074273092921297952, 6.84064873815925500996451684206, 8.157821683798138923936886508683, 9.168928442540544782523503572020, 9.607451970572798908743601803746
