| L(s) = 1 | + 0.327·2-s − 1.21·3-s − 1.89·4-s + 2.01·5-s − 0.396·6-s + 0.699·7-s − 1.27·8-s − 1.53·9-s + 0.660·10-s − 4.61·11-s + 2.29·12-s + 0.228·14-s − 2.44·15-s + 3.36·16-s + 4.66·17-s − 0.500·18-s + 2.12·19-s − 3.81·20-s − 0.847·21-s − 1.51·22-s + 0.894·23-s + 1.54·24-s − 0.928·25-s + 5.49·27-s − 1.32·28-s − 3.00·29-s − 0.800·30-s + ⋯ |
| L(s) = 1 | + 0.231·2-s − 0.699·3-s − 0.946·4-s + 0.902·5-s − 0.161·6-s + 0.264·7-s − 0.450·8-s − 0.510·9-s + 0.208·10-s − 1.39·11-s + 0.662·12-s + 0.0611·14-s − 0.631·15-s + 0.842·16-s + 1.13·17-s − 0.118·18-s + 0.486·19-s − 0.854·20-s − 0.184·21-s − 0.322·22-s + 0.186·23-s + 0.315·24-s − 0.185·25-s + 1.05·27-s − 0.250·28-s − 0.557·29-s − 0.146·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.327T + 2T^{2} \) |
| 3 | \( 1 + 1.21T + 3T^{2} \) |
| 5 | \( 1 - 2.01T + 5T^{2} \) |
| 7 | \( 1 - 0.699T + 7T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 17 | \( 1 - 4.66T + 17T^{2} \) |
| 19 | \( 1 - 2.12T + 19T^{2} \) |
| 23 | \( 1 - 0.894T + 23T^{2} \) |
| 29 | \( 1 + 3.00T + 29T^{2} \) |
| 37 | \( 1 + 9.14T + 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 - 0.0731T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 6.80T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 1.63T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 7.88T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 + 5.02T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−7.893279081618026910872749608306, −7.14444368161505890775127737114, −5.92643785769307164859884976742, −5.51344815378263338979613201977, −5.29079080446152084665050235304, −4.32952741529923187763340394994, −3.26513957932996864291438107333, −2.48093438038815901733320835536, −1.16009418575403076540459672906, 0,
1.16009418575403076540459672906, 2.48093438038815901733320835536, 3.26513957932996864291438107333, 4.32952741529923187763340394994, 5.29079080446152084665050235304, 5.51344815378263338979613201977, 5.92643785769307164859884976742, 7.14444368161505890775127737114, 7.893279081618026910872749608306
