| L(s) = 1 | + 2-s − 3.17·3-s + 4-s + 5-s − 3.17·6-s − 3.34·7-s + 8-s + 7.11·9-s + 10-s + 6.18·11-s − 3.17·12-s + 0.738·13-s − 3.34·14-s − 3.17·15-s + 16-s + 0.868·17-s + 7.11·18-s + 20-s + 10.6·21-s + 6.18·22-s − 1.14·23-s − 3.17·24-s + 25-s + 0.738·26-s − 13.0·27-s − 3.34·28-s + 7.78·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.83·3-s + 0.5·4-s + 0.447·5-s − 1.29·6-s − 1.26·7-s + 0.353·8-s + 2.37·9-s + 0.316·10-s + 1.86·11-s − 0.917·12-s + 0.204·13-s − 0.894·14-s − 0.820·15-s + 0.250·16-s + 0.210·17-s + 1.67·18-s + 0.223·20-s + 2.32·21-s + 1.31·22-s − 0.238·23-s − 0.649·24-s + 0.200·25-s + 0.144·26-s − 2.51·27-s − 0.632·28-s + 1.44·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.657394757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657394757\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 3.17T + 3T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 - 6.18T + 11T^{2} \) |
| 13 | \( 1 - 0.738T + 13T^{2} \) |
| 17 | \( 1 - 0.868T + 17T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 - 7.78T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 + 9.69T + 41T^{2} \) |
| 43 | \( 1 - 1.85T + 43T^{2} \) |
| 47 | \( 1 - 0.331T + 47T^{2} \) |
| 53 | \( 1 - 7.10T + 53T^{2} \) |
| 59 | \( 1 + 4.86T + 59T^{2} \) |
| 61 | \( 1 - 7.22T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 - 6.89T + 71T^{2} \) |
| 73 | \( 1 - 7.81T + 73T^{2} \) |
| 79 | \( 1 - 2.29T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 2.32T + 89T^{2} \) |
| 97 | \( 1 - 6.84T + 97T^{2} \) |
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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
−8.683064494204646054674179294142, −7.14701738666445762167970274860, −6.72041800989729015287062465916, −6.24038723552805218317090762288, −5.72233189496571030218833500779, −4.88396119994378429436783462402, −4.03532344107847077620656497840, −3.36093837338328579668822178724, −1.78383145341501670933185078319, −0.77299859370099034304740552103,
0.77299859370099034304740552103, 1.78383145341501670933185078319, 3.36093837338328579668822178724, 4.03532344107847077620656497840, 4.88396119994378429436783462402, 5.72233189496571030218833500779, 6.24038723552805218317090762288, 6.72041800989729015287062465916, 7.14701738666445762167970274860, 8.683064494204646054674179294142
