| L(s) = 1 | + 2-s − 0.482·3-s + 4-s + 5-s − 0.482·6-s − 0.948·7-s + 8-s − 2.76·9-s + 10-s − 1.63·11-s − 0.482·12-s − 13-s − 0.948·14-s − 0.482·15-s + 16-s − 5.37·17-s − 2.76·18-s + 4.68·19-s + 20-s + 0.458·21-s − 1.63·22-s + 7.75·23-s − 0.482·24-s + 25-s − 26-s + 2.78·27-s − 0.948·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.278·3-s + 0.5·4-s + 0.447·5-s − 0.197·6-s − 0.358·7-s + 0.353·8-s − 0.922·9-s + 0.316·10-s − 0.493·11-s − 0.139·12-s − 0.277·13-s − 0.253·14-s − 0.124·15-s + 0.250·16-s − 1.30·17-s − 0.652·18-s + 1.07·19-s + 0.223·20-s + 0.0999·21-s − 0.348·22-s + 1.61·23-s − 0.0985·24-s + 0.200·25-s − 0.196·26-s + 0.535·27-s − 0.179·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 0.482T + 3T^{2} \) |
| 7 | \( 1 + 0.948T + 7T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 - 7.75T + 23T^{2} \) |
| 29 | \( 1 + 2.02T + 29T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 + 0.897T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 - 8.67T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 3.40T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 4.69T + 83T^{2} \) |
| 89 | \( 1 - 1.42T + 89T^{2} \) |
| 97 | \( 1 + 6.42T + 97T^{2} \) |
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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
−7.998820977699132633918410078561, −7.01687616374260679830891257564, −6.62389930868375690288712631589, −5.60503666606203127036431666111, −5.24946378001103675400640450040, −4.44581292051944433179377149970, −3.18146963348243753330067269368, −2.79157255272415169651986944533, −1.60323339548256860405700365381, 0,
1.60323339548256860405700365381, 2.79157255272415169651986944533, 3.18146963348243753330067269368, 4.44581292051944433179377149970, 5.24946378001103675400640450040, 5.60503666606203127036431666111, 6.62389930868375690288712631589, 7.01687616374260679830891257564, 7.998820977699132633918410078561
