L(s) = 1 | + 2-s + 1.10·3-s + 4-s + 5-s + 1.10·6-s + 4.92·7-s + 8-s − 1.76·9-s + 10-s + 4.41·11-s + 1.10·12-s − 2.63·13-s + 4.92·14-s + 1.10·15-s + 16-s + 5.02·17-s − 1.76·18-s + 20-s + 5.46·21-s + 4.41·22-s − 1.02·23-s + 1.10·24-s + 25-s − 2.63·26-s − 5.29·27-s + 4.92·28-s − 2.66·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.640·3-s + 0.5·4-s + 0.447·5-s + 0.452·6-s + 1.86·7-s + 0.353·8-s − 0.589·9-s + 0.316·10-s + 1.33·11-s + 0.320·12-s − 0.732·13-s + 1.31·14-s + 0.286·15-s + 0.250·16-s + 1.21·17-s − 0.417·18-s + 0.223·20-s + 1.19·21-s + 0.940·22-s − 0.214·23-s + 0.226·24-s + 0.200·25-s − 0.517·26-s − 1.01·27-s + 0.931·28-s − 0.494·29-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
\(L(1)\) |
\(\approx\) |
\(5.223958229\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.223958229\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.10T + 3T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 - 4.41T + 11T^{2} \) |
| 13 | \( 1 + 2.63T + 13T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 23 | \( 1 + 1.02T + 23T^{2} \) |
| 29 | \( 1 + 2.66T + 29T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + 2.00T + 37T^{2} \) |
| 41 | \( 1 + 2.15T + 41T^{2} \) |
| 43 | \( 1 - 1.51T + 43T^{2} \) |
| 47 | \( 1 - 1.79T + 47T^{2} \) |
| 53 | \( 1 + 5.62T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 2.83T + 79T^{2} \) |
| 83 | \( 1 + 7.28T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.437802137636297717690745157628, −7.79475412562422498145905297168, −7.21142075059848883391937819246, −6.10222713298801701005492353009, −5.43887525436147778415112331371, −4.76127389231018844468331003387, −3.93460408352054707317572315955, −3.04612731792552559395269908145, −2.00971495385049747604006197841, −1.39252294996221243510332203796,
1.39252294996221243510332203796, 2.00971495385049747604006197841, 3.04612731792552559395269908145, 3.93460408352054707317572315955, 4.76127389231018844468331003387, 5.43887525436147778415112331371, 6.10222713298801701005492353009, 7.21142075059848883391937819246, 7.79475412562422498145905297168, 8.437802137636297717690745157628