| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 366.·5-s + 216·6-s − 561.·7-s + 512·8-s + 729·9-s + 2.93e3·10-s − 1.09e3·11-s + 1.72e3·12-s + 6.92e3·13-s − 4.49e3·14-s + 9.90e3·15-s + 4.09e3·16-s + 1.36e4·17-s + 5.83e3·18-s + 2.95e4·19-s + 2.34e4·20-s − 1.51e4·21-s − 8.76e3·22-s − 1.21e4·23-s + 1.38e4·24-s + 5.64e4·25-s + 5.54e4·26-s + 1.96e4·27-s − 3.59e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.31·5-s + 0.408·6-s − 0.618·7-s + 0.353·8-s + 0.333·9-s + 0.928·10-s − 0.248·11-s + 0.288·12-s + 0.874·13-s − 0.437·14-s + 0.757·15-s + 0.250·16-s + 0.674·17-s + 0.235·18-s + 0.988·19-s + 0.656·20-s − 0.357·21-s − 0.175·22-s − 0.208·23-s + 0.204·24-s + 0.722·25-s + 0.618·26-s + 0.192·27-s − 0.309·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.164107419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.164107419\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 5 | \( 1 - 366.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 561.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.09e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.36e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.95e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.50e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.04e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.92e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.71e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.30e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.28e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.95e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.46e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.82e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.03e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.36e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.27e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.88e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.46e6T + 8.07e13T^{2} \) |
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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
−12.18081552014869323738602136870, −10.67303609987841173162572361520, −9.830488269575078325427696370905, −8.857144907892289348902863686881, −7.38246247312100045202263473206, −6.17555348447449859871475694932, −5.34355011466900104827637636938, −3.67277670272364354873462432855, −2.59994987819479963386236607898, −1.31951532557033063465385384888,
1.31951532557033063465385384888, 2.59994987819479963386236607898, 3.67277670272364354873462432855, 5.34355011466900104827637636938, 6.17555348447449859871475694932, 7.38246247312100045202263473206, 8.857144907892289348902863686881, 9.830488269575078325427696370905, 10.67303609987841173162572361520, 12.18081552014869323738602136870
