| L(s) = 1 | − 0.381·2-s + 3-s − 1.85·4-s − 2.23·5-s − 0.381·6-s + 1.47·8-s + 9-s + 0.854·10-s − 11-s − 1.85·12-s + 5.47·13-s − 2.23·15-s + 3.14·16-s − 6·17-s − 0.381·18-s − 0.236·19-s + 4.14·20-s + 0.381·22-s + 6.47·23-s + 1.47·24-s − 2.09·26-s + 27-s − 5.76·29-s + 0.854·30-s + 0.472·31-s − 4.14·32-s − 33-s + ⋯ |
| L(s) = 1 | − 0.270·2-s + 0.577·3-s − 0.927·4-s − 0.999·5-s − 0.155·6-s + 0.520·8-s + 0.333·9-s + 0.270·10-s − 0.301·11-s − 0.535·12-s + 1.51·13-s − 0.577·15-s + 0.786·16-s − 1.45·17-s − 0.0900·18-s − 0.0541·19-s + 0.927·20-s + 0.0814·22-s + 1.34·23-s + 0.300·24-s − 0.409·26-s + 0.192·27-s − 1.07·29-s + 0.155·30-s + 0.0847·31-s − 0.732·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 0.236T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 5.76T + 29T^{2} \) |
| 31 | \( 1 - 0.472T + 31T^{2} \) |
| 37 | \( 1 + 9.47T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 - 4.94T + 53T^{2} \) |
| 59 | \( 1 + 5.94T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 6.47T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.807030341900377071986106821698, −8.458543213076420806943974649736, −7.60782286100225971478383860507, −6.84612894136881739739190128481, −5.62692514355561985503532827542, −4.54919738362808015152712971308, −3.92670638612145572212418942778, −3.12806557848555480053285195221, −1.51088916595908606697405159221, 0,
1.51088916595908606697405159221, 3.12806557848555480053285195221, 3.92670638612145572212418942778, 4.54919738362808015152712971308, 5.62692514355561985503532827542, 6.84612894136881739739190128481, 7.60782286100225971478383860507, 8.458543213076420806943974649736, 8.807030341900377071986106821698
