| L(s) = 1 | − 4.89·2-s + 6.57·3-s + 15.9·4-s + 15.5·5-s − 32.1·6-s − 7·7-s − 38.7·8-s + 16.1·9-s − 76.0·10-s − 11·11-s + 104.·12-s + 74.3·13-s + 34.2·14-s + 102.·15-s + 62.1·16-s − 94.0·17-s − 79.2·18-s + 135.·19-s + 247.·20-s − 46.0·21-s + 53.8·22-s + 81.1·23-s − 254.·24-s + 116.·25-s − 363.·26-s − 70.9·27-s − 111.·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.26·3-s + 1.99·4-s + 1.39·5-s − 2.18·6-s − 0.377·7-s − 1.71·8-s + 0.599·9-s − 2.40·10-s − 0.301·11-s + 2.51·12-s + 1.58·13-s + 0.653·14-s + 1.75·15-s + 0.970·16-s − 1.34·17-s − 1.03·18-s + 1.63·19-s + 2.76·20-s − 0.478·21-s + 0.521·22-s + 0.735·23-s − 2.16·24-s + 0.934·25-s − 2.74·26-s − 0.506·27-s − 0.752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.222417095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222417095\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.89T + 8T^{2} \) |
| 3 | \( 1 - 6.57T + 27T^{2} \) |
| 5 | \( 1 - 15.5T + 125T^{2} \) |
| 13 | \( 1 - 74.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 81.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 53.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.50T + 2.97e4T^{2} \) |
| 37 | \( 1 + 9.14T + 5.06e4T^{2} \) |
| 41 | \( 1 + 339.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 433.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 54.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 534.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 358.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 694.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 278.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 886.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 185.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 122.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 847.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−13.80709860670809673288055522489, −13.31430364245378899155340339891, −11.20558046210743097284177633725, −10.13980698243636919185798125497, −9.136415183947955875825588239580, −8.796999221867646801378178713084, −7.40839495440179040815122085971, −6.07986757278846757743002364832, −2.91872760787979495632881567677, −1.58216166679708247101339440373,
1.58216166679708247101339440373, 2.91872760787979495632881567677, 6.07986757278846757743002364832, 7.40839495440179040815122085971, 8.796999221867646801378178713084, 9.136415183947955875825588239580, 10.13980698243636919185798125497, 11.20558046210743097284177633725, 13.31430364245378899155340339891, 13.80709860670809673288055522489
