| L(s) = 1 | + 0.351·2-s + 3.22·3-s − 7.87·4-s + 13.1·5-s + 1.13·6-s + 33.2·7-s − 5.57·8-s − 16.5·9-s + 4.62·10-s + 13.6·11-s − 25.4·12-s + 5.12·13-s + 11.6·14-s + 42.5·15-s + 61.0·16-s − 5.82·18-s − 38.4·19-s − 103.·20-s + 107.·21-s + 4.78·22-s + 18.7·23-s − 17.9·24-s + 48.9·25-s + 1.80·26-s − 140.·27-s − 262.·28-s + 211.·29-s + ⋯ |
| L(s) = 1 | + 0.124·2-s + 0.621·3-s − 0.984·4-s + 1.17·5-s + 0.0770·6-s + 1.79·7-s − 0.246·8-s − 0.614·9-s + 0.146·10-s + 0.373·11-s − 0.611·12-s + 0.109·13-s + 0.223·14-s + 0.732·15-s + 0.954·16-s − 0.0762·18-s − 0.463·19-s − 1.16·20-s + 1.11·21-s + 0.0463·22-s + 0.169·23-s − 0.152·24-s + 0.391·25-s + 0.0135·26-s − 1.00·27-s − 1.76·28-s + 1.35·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.774507038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.774507038\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 0.351T + 8T^{2} \) |
| 3 | \( 1 - 3.22T + 27T^{2} \) |
| 5 | \( 1 - 13.1T + 125T^{2} \) |
| 7 | \( 1 - 33.2T + 343T^{2} \) |
| 11 | \( 1 - 13.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.12T + 2.19e3T^{2} \) |
| 19 | \( 1 + 38.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 211.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 234.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 2.77T + 5.06e4T^{2} \) |
| 41 | \( 1 - 303.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 149.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 366.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 499.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 507.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 33.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 442.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 475.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 214.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 325.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 240.T + 9.12e5T^{2} \) |
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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
−11.33882693752712061295240580302, −10.29562897972344570596342367138, −9.280916059205480459473415846877, −8.547174628801394398053237904399, −7.924218376786530949654221549884, −6.15414723862678520521118566035, −5.17600641403783846393419526063, −4.28706706614943452240271448813, −2.60346920895574014259434950527, −1.29550476470275882825290789027,
1.29550476470275882825290789027, 2.60346920895574014259434950527, 4.28706706614943452240271448813, 5.17600641403783846393419526063, 6.15414723862678520521118566035, 7.924218376786530949654221549884, 8.547174628801394398053237904399, 9.280916059205480459473415846877, 10.29562897972344570596342367138, 11.33882693752712061295240580302
