| L(s) = 1 | + 2.73·2-s − 3-s + 5.46·4-s − 5-s − 2.73·6-s + 7-s + 9.46·8-s + 9-s − 2.73·10-s − 11-s − 5.46·12-s + 6.19·13-s + 2.73·14-s + 15-s + 14.9·16-s + 0.464·17-s + 2.73·18-s − 8.46·19-s − 5.46·20-s − 21-s − 2.73·22-s + 1.53·23-s − 9.46·24-s + 25-s + 16.9·26-s − 27-s + 5.46·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.577·3-s + 2.73·4-s − 0.447·5-s − 1.11·6-s + 0.377·7-s + 3.34·8-s + 0.333·9-s − 0.863·10-s − 0.301·11-s − 1.57·12-s + 1.71·13-s + 0.730·14-s + 0.258·15-s + 3.73·16-s + 0.112·17-s + 0.643·18-s − 1.94·19-s − 1.22·20-s − 0.218·21-s − 0.582·22-s + 0.320·23-s − 1.93·24-s + 0.200·25-s + 3.31·26-s − 0.192·27-s + 1.03·28-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
| \(L(1)\) |
\(\approx\) |
\(4.659020815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.659020815\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.73T + 2T^{2} \) |
| 13 | \( 1 - 6.19T + 13T^{2} \) |
| 17 | \( 1 - 0.464T + 17T^{2} \) |
| 19 | \( 1 + 8.46T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 - 2.26T + 29T^{2} \) |
| 31 | \( 1 + 8.73T + 31T^{2} \) |
| 37 | \( 1 - 6.19T + 37T^{2} \) |
| 41 | \( 1 - 9.66T + 41T^{2} \) |
| 43 | \( 1 + 1.73T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 + 2.46T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 9.39T + 61T^{2} \) |
| 67 | \( 1 + 8.92T + 67T^{2} \) |
| 71 | \( 1 + 9.66T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 4.46T + 83T^{2} \) |
| 89 | \( 1 + 2.66T + 89T^{2} \) |
| 97 | \( 1 + 5.73T + 97T^{2} \) |
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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
−10.53058924060965135279181518257, −8.772727463393138777089024584474, −7.79463769549596060507922757008, −6.91388055414940181746465443756, −6.06736263841106148280350070091, −5.60135403566506504430228925351, −4.35677293668797325777684366300, −4.07479067432813116528226032217, −2.84823814425087349508888438448, −1.55077423004073157023590535648,
1.55077423004073157023590535648, 2.84823814425087349508888438448, 4.07479067432813116528226032217, 4.35677293668797325777684366300, 5.60135403566506504430228925351, 6.06736263841106148280350070091, 6.91388055414940181746465443756, 7.79463769549596060507922757008, 8.772727463393138777089024584474, 10.53058924060965135279181518257
