| L(s) = 1 | − 2-s − 0.246·3-s + 4-s − 2.06·5-s + 0.246·6-s + 1.96·7-s − 8-s − 2.93·9-s + 2.06·10-s + 2.64·11-s − 0.246·12-s + 2.31·13-s − 1.96·14-s + 0.510·15-s + 16-s + 7.06·17-s + 2.93·18-s + 1.73·19-s − 2.06·20-s − 0.484·21-s − 2.64·22-s + 3.74·23-s + 0.246·24-s − 0.721·25-s − 2.31·26-s + 1.46·27-s + 1.96·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.142·3-s + 0.5·4-s − 0.925·5-s + 0.100·6-s + 0.742·7-s − 0.353·8-s − 0.979·9-s + 0.654·10-s + 0.796·11-s − 0.0711·12-s + 0.641·13-s − 0.524·14-s + 0.131·15-s + 0.250·16-s + 1.71·17-s + 0.692·18-s + 0.397·19-s − 0.462·20-s − 0.105·21-s − 0.563·22-s + 0.781·23-s + 0.0503·24-s − 0.144·25-s − 0.453·26-s + 0.281·27-s + 0.371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.178160435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178160435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 0.246T + 3T^{2} \) |
| 5 | \( 1 + 2.06T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 - 7.06T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 9.87T + 37T^{2} \) |
| 41 | \( 1 + 3.07T + 41T^{2} \) |
| 43 | \( 1 - 8.06T + 43T^{2} \) |
| 47 | \( 1 + 1.25T + 47T^{2} \) |
| 53 | \( 1 + 7.20T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 0.897T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 3.84T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−8.374895053409544542273903145592, −7.71791592464423493656111357215, −7.38865108480827379376647983892, −6.17496580672470569986630548671, −5.66587163655201856184811628471, −4.67954817822873023413476811534, −3.63517663503707443222189639488, −3.07135227747934515767676887442, −1.65478414363612373801450911928, −0.73295703075847911410947204567,
0.73295703075847911410947204567, 1.65478414363612373801450911928, 3.07135227747934515767676887442, 3.63517663503707443222189639488, 4.67954817822873023413476811534, 5.66587163655201856184811628471, 6.17496580672470569986630548671, 7.38865108480827379376647983892, 7.71791592464423493656111357215, 8.374895053409544542273903145592
