| L(s) = 1 | + 15.3·2-s − 27·3-s + 108.·4-s − 358.·5-s − 415.·6-s − 482.·7-s − 297.·8-s + 729·9-s − 5.51e3·10-s − 2.93e3·12-s + 2.67e3·13-s − 7.41e3·14-s + 9.67e3·15-s − 1.84e4·16-s − 2.89e4·17-s + 1.12e4·18-s − 5.12e4·19-s − 3.89e4·20-s + 1.30e4·21-s + 6.57e4·23-s + 8.03e3·24-s + 5.03e4·25-s + 4.11e4·26-s − 1.96e4·27-s − 5.24e4·28-s − 1.49e5·29-s + 1.48e5·30-s + ⋯ |
| L(s) = 1 | + 1.35·2-s − 0.577·3-s + 0.848·4-s − 1.28·5-s − 0.785·6-s − 0.531·7-s − 0.205·8-s + 0.333·9-s − 1.74·10-s − 0.490·12-s + 0.338·13-s − 0.722·14-s + 0.740·15-s − 1.12·16-s − 1.42·17-s + 0.453·18-s − 1.71·19-s − 1.08·20-s + 0.306·21-s + 1.12·23-s + 0.118·24-s + 0.644·25-s + 0.459·26-s − 0.192·27-s − 0.451·28-s − 1.13·29-s + 1.00·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.137085166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.137085166\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 15.3T + 128T^{2} \) |
| 5 | \( 1 + 358.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 482.T + 8.23e5T^{2} \) |
| 13 | \( 1 - 2.67e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.89e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.12e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.57e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.49e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.03e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.49e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.60e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.13e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.09e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.53e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.36e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.58e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.99e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 9.01e4T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.24e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.04e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.12e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.51e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.05e7T + 8.07e13T^{2} \) |
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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
−10.86481288021333085273493083992, −9.251199783670701974361798378149, −8.283483703223598299755161746680, −6.89049484442831233424088722979, −6.39918085487730772517312813134, −5.15477104136742332244457399666, −4.23075725589110267263696611743, −3.68685876760292194183127524342, −2.37276743985865646307103717762, −0.39612401578055602986691127361,
0.39612401578055602986691127361, 2.37276743985865646307103717762, 3.68685876760292194183127524342, 4.23075725589110267263696611743, 5.15477104136742332244457399666, 6.39918085487730772517312813134, 6.89049484442831233424088722979, 8.283483703223598299755161746680, 9.251199783670701974361798378149, 10.86481288021333085273493083992
