| L(s) = 1 | − 11.5·2-s + 6.48·4-s + 401.·5-s + 213.·7-s + 1.40e3·8-s − 4.65e3·10-s + 4.97e3·11-s − 1.26e4·13-s − 2.47e3·14-s − 1.71e4·16-s − 3.69e4·17-s − 3.10e4·19-s + 2.60e3·20-s − 5.76e4·22-s − 5.25e4·23-s + 8.27e4·25-s + 1.46e5·26-s + 1.38e3·28-s + 2.09e5·29-s − 1.18e4·31-s + 1.87e4·32-s + 4.28e5·34-s + 8.56e4·35-s + 1.69e4·37-s + 3.60e5·38-s + 5.65e5·40-s − 8.01e5·41-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.0506·4-s + 1.43·5-s + 0.235·7-s + 0.973·8-s − 1.47·10-s + 1.12·11-s − 1.59·13-s − 0.241·14-s − 1.04·16-s − 1.82·17-s − 1.03·19-s + 0.0726·20-s − 1.15·22-s − 0.901·23-s + 1.05·25-s + 1.63·26-s + 0.0119·28-s + 1.59·29-s − 0.0715·31-s + 0.101·32-s + 1.86·34-s + 0.337·35-s + 0.0549·37-s + 1.06·38-s + 1.39·40-s − 1.81·41-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
| \(L(4)\) |
\(\approx\) |
\(1.224933961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224933961\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 11.5T + 128T^{2} \) |
| 5 | \( 1 - 401.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 213.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.97e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.26e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.69e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.10e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.25e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.18e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.69e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.01e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.20e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.44e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.95e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.37e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.05e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.70e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.65e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.37e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.08e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.42e6T + 8.07e13T^{2} \) |
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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
−9.689820091966241185866940145407, −8.951979791597900118165581933858, −8.273257609017840694939428113715, −6.92057186317757066190479335299, −6.39366238964314588776959752514, −5.00346473002792634557084047611, −4.27465221319148705610791599159, −2.28449956520484317956107506087, −1.85493606853484406698066522994, −0.55586962972360250904526085203,
0.55586962972360250904526085203, 1.85493606853484406698066522994, 2.28449956520484317956107506087, 4.27465221319148705610791599159, 5.00346473002792634557084047611, 6.39366238964314588776959752514, 6.92057186317757066190479335299, 8.273257609017840694939428113715, 8.951979791597900118165581933858, 9.689820091966241185866940145407
