| L(s) = 1 | + 6.85·2-s − 81.0·4-s + 190.·5-s − 799.·7-s − 1.43e3·8-s + 1.30e3·10-s − 972.·11-s − 1.27e3·13-s − 5.47e3·14-s + 552.·16-s − 3.64e4·17-s + 1.90e3·19-s − 1.54e4·20-s − 6.66e3·22-s − 8.01e4·23-s − 4.17e4·25-s − 8.76e3·26-s + 6.47e4·28-s + 2.01e5·29-s + 1.06e5·31-s + 1.87e5·32-s − 2.49e5·34-s − 1.52e5·35-s − 2.46e5·37-s + 1.30e4·38-s − 2.73e5·40-s + 3.48e5·41-s + ⋯ |
| L(s) = 1 | + 0.605·2-s − 0.633·4-s + 0.682·5-s − 0.880·7-s − 0.989·8-s + 0.413·10-s − 0.220·11-s − 0.161·13-s − 0.533·14-s + 0.0337·16-s − 1.80·17-s + 0.0635·19-s − 0.431·20-s − 0.133·22-s − 1.37·23-s − 0.534·25-s − 0.0977·26-s + 0.557·28-s + 1.53·29-s + 0.644·31-s + 1.00·32-s − 1.09·34-s − 0.601·35-s − 0.799·37-s + 0.0385·38-s − 0.675·40-s + 0.789·41-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
| \(L(4)\) |
\(\approx\) |
\(1.207956383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207956383\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 6.85T + 128T^{2} \) |
| 5 | \( 1 - 190.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 799.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 972.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.27e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.64e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.90e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.01e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.01e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.06e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.46e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.48e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.03e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.12e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.34e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.14e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.58e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.94e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.28e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.63e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.34e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.79e6T + 8.07e13T^{2} \) |
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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
−9.700743882952948073924175518903, −8.971804429669864271198613340804, −8.044880872772624717835411127286, −6.51655419276751952288986723519, −6.13892892661367122425689806413, −4.96403221781816114759650708021, −4.16661085887546211813424134080, −3.05460004074832494495345738618, −2.06772172297438085841950911914, −0.41383904490772805646686009437,
0.41383904490772805646686009437, 2.06772172297438085841950911914, 3.05460004074832494495345738618, 4.16661085887546211813424134080, 4.96403221781816114759650708021, 6.13892892661367122425689806413, 6.51655419276751952288986723519, 8.044880872772624717835411127286, 8.971804429669864271198613340804, 9.700743882952948073924175518903
