| L(s) = 1 | − 2.04·2-s + 2.19·4-s − 3.85·5-s + 4.47·7-s − 0.408·8-s + 7.89·10-s − 11-s + 3.84·13-s − 9.17·14-s − 3.56·16-s − 2.20·17-s − 6.72·19-s − 8.47·20-s + 2.04·22-s + 6.81·23-s + 9.84·25-s − 7.87·26-s + 9.84·28-s − 4.03·29-s + 1.44·31-s + 8.11·32-s + 4.50·34-s − 17.2·35-s − 2.37·37-s + 13.7·38-s + 1.57·40-s + 5.71·41-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.09·4-s − 1.72·5-s + 1.69·7-s − 0.144·8-s + 2.49·10-s − 0.301·11-s + 1.06·13-s − 2.45·14-s − 0.890·16-s − 0.533·17-s − 1.54·19-s − 1.89·20-s + 0.436·22-s + 1.42·23-s + 1.96·25-s − 1.54·26-s + 1.86·28-s − 0.748·29-s + 0.259·31-s + 1.43·32-s + 0.773·34-s − 2.91·35-s − 0.390·37-s + 2.23·38-s + 0.248·40-s + 0.892·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 13 | \( 1 - 3.84T + 13T^{2} \) |
| 17 | \( 1 + 2.20T + 17T^{2} \) |
| 19 | \( 1 + 6.72T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 + 4.03T + 29T^{2} \) |
| 31 | \( 1 - 1.44T + 31T^{2} \) |
| 37 | \( 1 + 2.37T + 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 - 3.94T + 43T^{2} \) |
| 47 | \( 1 + 8.31T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 7.39T + 59T^{2} \) |
| 61 | \( 1 - 0.0871T + 61T^{2} \) |
| 67 | \( 1 + 3.72T + 67T^{2} \) |
| 71 | \( 1 - 0.424T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.70754622435584023844314287863, −7.24737626110460601330433549126, −6.50239292492048130759109603877, −5.30205770890380973493559401609, −4.32295630132238847286324249443, −4.23775206583667072623526525101, −2.91409838219690370283294824415, −1.82034706096290204973789726762, −1.03693871661550962234038135554, 0,
1.03693871661550962234038135554, 1.82034706096290204973789726762, 2.91409838219690370283294824415, 4.23775206583667072623526525101, 4.32295630132238847286324249443, 5.30205770890380973493559401609, 6.50239292492048130759109603877, 7.24737626110460601330433549126, 7.70754622435584023844314287863
