| L(s) = 1 | − 8·2-s − 82.2·3-s + 64·4-s − 99.7·5-s + 657.·6-s + 343·7-s − 512·8-s + 4.57e3·9-s + 798.·10-s + 756.·11-s − 5.26e3·12-s + 2.19e3·13-s − 2.74e3·14-s + 8.20e3·15-s + 4.09e3·16-s + 2.87e4·17-s − 3.66e4·18-s + 1.22e3·19-s − 6.38e3·20-s − 2.82e4·21-s − 6.05e3·22-s − 4.67e4·23-s + 4.21e4·24-s − 6.81e4·25-s − 1.75e4·26-s − 1.96e5·27-s + 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.75·3-s + 0.5·4-s − 0.357·5-s + 1.24·6-s + 0.377·7-s − 0.353·8-s + 2.09·9-s + 0.252·10-s + 0.171·11-s − 0.879·12-s + 0.277·13-s − 0.267·14-s + 0.627·15-s + 0.250·16-s + 1.41·17-s − 1.47·18-s + 0.0409·19-s − 0.178·20-s − 0.664·21-s − 0.121·22-s − 0.800·23-s + 0.621·24-s − 0.872·25-s − 0.196·26-s − 1.92·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6006876715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6006876715\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 3 | \( 1 + 82.2T + 2.18e3T^{2} \) |
| 5 | \( 1 + 99.7T + 7.81e4T^{2} \) |
| 11 | \( 1 - 756.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.87e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.22e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.67e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.28e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.67e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.03e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.89e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.46e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.18e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.40e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.31e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.75e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.25e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.90e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.03e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.17e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.96e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.74e6T + 8.07e13T^{2} \) |
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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
−11.39300980870211015587560168387, −10.46067622119473706093270595191, −9.676251612634452023920311049564, −8.120415487584853607861412461098, −7.18384520248621344491549359879, −6.06580562713550526021749857081, −5.24802158905732718408917326972, −3.83212427765339496377729591893, −1.61386082693196677406269113260, −0.53858984527119431464281845963,
0.53858984527119431464281845963, 1.61386082693196677406269113260, 3.83212427765339496377729591893, 5.24802158905732718408917326972, 6.06580562713550526021749857081, 7.18384520248621344491549359879, 8.120415487584853607861412461098, 9.676251612634452023920311049564, 10.46067622119473706093270595191, 11.39300980870211015587560168387
