| L(s) = 1 | + 13.3·2-s + 49.7·4-s − 152.·5-s − 1.32e3·7-s − 1.04e3·8-s − 2.02e3·10-s − 5.13e3·11-s − 1.22e4·13-s − 1.77e4·14-s − 2.02e4·16-s − 2.66e4·17-s + 3.31e4·19-s − 7.57e3·20-s − 6.84e4·22-s + 9.70e4·23-s − 5.49e4·25-s − 1.63e5·26-s − 6.61e4·28-s − 2.09e5·29-s − 3.08e5·31-s − 1.36e5·32-s − 3.54e5·34-s + 2.02e5·35-s + 2.87e5·37-s + 4.42e5·38-s + 1.58e5·40-s − 5.81e5·41-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.388·4-s − 0.544·5-s − 1.46·7-s − 0.720·8-s − 0.641·10-s − 1.16·11-s − 1.54·13-s − 1.72·14-s − 1.23·16-s − 1.31·17-s + 1.11·19-s − 0.211·20-s − 1.37·22-s + 1.66·23-s − 0.703·25-s − 1.82·26-s − 0.569·28-s − 1.59·29-s − 1.86·31-s − 0.738·32-s − 1.54·34-s + 0.796·35-s + 0.934·37-s + 1.30·38-s + 0.391·40-s − 1.31·41-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
| \(L(4)\) |
\(\approx\) |
\(0.09512154151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09512154151\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 13.3T + 128T^{2} \) |
| 5 | \( 1 + 152.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.32e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.13e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.22e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.66e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.31e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.70e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.08e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.87e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.81e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.40e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.50e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.99e4T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.00e4T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.38e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.24e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.83e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.62e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.79e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.62e7T + 8.07e13T^{2} \) |
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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
−9.531909527002019579693875736022, −9.119425308816619775585435324231, −7.48170331527492928418634019961, −7.00248971487258964441383027987, −5.72147205119177334726806649731, −5.08412050423020331776478187567, −4.02675971216865465469768690726, −3.11824407360470971125880413123, −2.43426443343464621244192430159, −0.10164685246247913541882971584,
0.10164685246247913541882971584, 2.43426443343464621244192430159, 3.11824407360470971125880413123, 4.02675971216865465469768690726, 5.08412050423020331776478187567, 5.72147205119177334726806649731, 7.00248971487258964441383027987, 7.48170331527492928418634019961, 9.119425308816619775585435324231, 9.531909527002019579693875736022
