| L(s) = 1 | + 0.846·2-s − 1.28·4-s + 1.31·5-s + 0.998·7-s − 2.77·8-s + 1.11·10-s + 2.32·11-s − 2.21·13-s + 0.845·14-s + 0.216·16-s + 7.02·17-s + 2.02·19-s − 1.69·20-s + 1.96·22-s − 6.86·23-s − 3.26·25-s − 1.87·26-s − 1.28·28-s + 8.13·29-s − 1.12·31-s + 5.74·32-s + 5.94·34-s + 1.31·35-s + 1.32·37-s + 1.71·38-s − 3.66·40-s + 0.958·41-s + ⋯ |
| L(s) = 1 | + 0.598·2-s − 0.641·4-s + 0.589·5-s + 0.377·7-s − 0.982·8-s + 0.352·10-s + 0.701·11-s − 0.615·13-s + 0.225·14-s + 0.0541·16-s + 1.70·17-s + 0.464·19-s − 0.378·20-s + 0.419·22-s − 1.43·23-s − 0.652·25-s − 0.368·26-s − 0.242·28-s + 1.51·29-s − 0.201·31-s + 1.01·32-s + 1.01·34-s + 0.222·35-s + 0.217·37-s + 0.277·38-s − 0.579·40-s + 0.149·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.351443312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.351443312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 - 0.846T + 2T^{2} \) |
| 5 | \( 1 - 1.31T + 5T^{2} \) |
| 7 | \( 1 - 0.998T + 7T^{2} \) |
| 11 | \( 1 - 2.32T + 11T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 - 7.02T + 17T^{2} \) |
| 19 | \( 1 - 2.02T + 19T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 - 8.13T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 - 1.32T + 37T^{2} \) |
| 41 | \( 1 - 0.958T + 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 - 5.98T + 59T^{2} \) |
| 61 | \( 1 + 3.49T + 61T^{2} \) |
| 67 | \( 1 - 5.32T + 67T^{2} \) |
| 71 | \( 1 + 0.783T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 - 6.41T + 79T^{2} \) |
| 83 | \( 1 + 0.452T + 83T^{2} \) |
| 89 | \( 1 + 2.32T + 89T^{2} \) |
| 97 | \( 1 + 7.13T + 97T^{2} \) |
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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
−9.175092445949854701642513267765, −8.281972889145513252926997976744, −7.62945446743483347534265455814, −6.48586091433378474643948257624, −5.70424251875505059086068637373, −5.17616982217540535106241035985, −4.19985346583907975393622154279, −3.46893375395391170629327476580, −2.31191186918366114801033428851, −0.968082966553721718908967466378,
0.968082966553721718908967466378, 2.31191186918366114801033428851, 3.46893375395391170629327476580, 4.19985346583907975393622154279, 5.17616982217540535106241035985, 5.70424251875505059086068637373, 6.48586091433378474643948257624, 7.62945446743483347534265455814, 8.281972889145513252926997976744, 9.175092445949854701642513267765
