| L(s) = 1 | + 2.34·2-s + 3-s + 3.48·4-s + 5-s + 2.34·6-s − 7-s + 3.48·8-s + 9-s + 2.34·10-s + 11-s + 3.48·12-s + 3.14·13-s − 2.34·14-s + 15-s + 1.19·16-s + 4.19·17-s + 2.34·18-s − 7.17·19-s + 3.48·20-s − 21-s + 2.34·22-s − 4.48·23-s + 3.48·24-s + 25-s + 7.37·26-s + 27-s − 3.48·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 0.577·3-s + 1.74·4-s + 0.447·5-s + 0.956·6-s − 0.377·7-s + 1.23·8-s + 0.333·9-s + 0.740·10-s + 0.301·11-s + 1.00·12-s + 0.872·13-s − 0.626·14-s + 0.258·15-s + 0.299·16-s + 1.01·17-s + 0.552·18-s − 1.64·19-s + 0.780·20-s − 0.218·21-s + 0.499·22-s − 0.936·23-s + 0.712·24-s + 0.200·25-s + 1.44·26-s + 0.192·27-s − 0.659·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.294665148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.294665148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 23 | \( 1 + 4.48T + 23T^{2} \) |
| 29 | \( 1 - 5.34T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 + 6.81T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 + 1.05T + 43T^{2} \) |
| 47 | \( 1 + 3.83T + 47T^{2} \) |
| 53 | \( 1 - 2.48T + 53T^{2} \) |
| 59 | \( 1 - 2.02T + 59T^{2} \) |
| 61 | \( 1 - 4.78T + 61T^{2} \) |
| 67 | \( 1 + 4.97T + 67T^{2} \) |
| 71 | \( 1 + 7.43T + 71T^{2} \) |
| 73 | \( 1 - 8.68T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 2.58T + 83T^{2} \) |
| 89 | \( 1 - 4.61T + 89T^{2} \) |
| 97 | \( 1 - 8.22T + 97T^{2} \) |
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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
−10.03562186622351731497998628099, −8.856721954676514406330169936037, −8.144936645314817696902072406783, −6.82700770885754097021968864876, −6.31666492078190340350377761070, −5.51703933870808662882755483247, −4.42658302705657473779611412073, −3.69417634297439624050722858446, −2.84260305159584516820937983815, −1.74846656866396397667162548706,
1.74846656866396397667162548706, 2.84260305159584516820937983815, 3.69417634297439624050722858446, 4.42658302705657473779611412073, 5.51703933870808662882755483247, 6.31666492078190340350377761070, 6.82700770885754097021968864876, 8.144936645314817696902072406783, 8.856721954676514406330169936037, 10.03562186622351731497998628099
