| L(s) = 1 | − 2.27·2-s + 3.42·3-s + 3.15·4-s − 0.765·5-s − 7.77·6-s − 2.63·8-s + 8.73·9-s + 1.73·10-s − 1.96·11-s + 10.8·12-s − 3.97·13-s − 2.62·15-s − 0.341·16-s − 1.82·17-s − 19.8·18-s + 4.34·19-s − 2.41·20-s + 4.45·22-s + 2.97·23-s − 9.01·24-s − 4.41·25-s + 9.02·26-s + 19.6·27-s − 2.01·29-s + 5.95·30-s + 9.52·31-s + 6.03·32-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.97·3-s + 1.57·4-s − 0.342·5-s − 3.17·6-s − 0.930·8-s + 2.91·9-s + 0.549·10-s − 0.592·11-s + 3.12·12-s − 1.10·13-s − 0.676·15-s − 0.0853·16-s − 0.443·17-s − 4.67·18-s + 0.996·19-s − 0.540·20-s + 0.950·22-s + 0.619·23-s − 1.83·24-s − 0.882·25-s + 1.76·26-s + 3.77·27-s − 0.373·29-s + 1.08·30-s + 1.71·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.570757460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570757460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 - 3.42T + 3T^{2} \) |
| 5 | \( 1 + 0.765T + 5T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 - 2.97T + 23T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 - 9.52T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 - 5.14T + 53T^{2} \) |
| 59 | \( 1 + 0.872T + 59T^{2} \) |
| 61 | \( 1 - 9.79T + 61T^{2} \) |
| 67 | \( 1 - 4.01T + 67T^{2} \) |
| 71 | \( 1 - 9.65T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 8.07T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−9.173010457096593591277748482224, −8.277864819985471098193523092670, −7.917854592684474589548290822489, −7.37476460296979806644923392408, −6.68420224712490379649234437744, −4.90560717518279370656106976370, −3.92707395736593435297798346505, −2.70114006483935736474300753187, −2.32266781127454364736136563402, −0.992043231654877471115082128693,
0.992043231654877471115082128693, 2.32266781127454364736136563402, 2.70114006483935736474300753187, 3.92707395736593435297798346505, 4.90560717518279370656106976370, 6.68420224712490379649234437744, 7.37476460296979806644923392408, 7.917854592684474589548290822489, 8.277864819985471098193523092670, 9.173010457096593591277748482224
