| L(s) = 1 | − 0.475·2-s − 1.77·4-s − 5-s − 2.21·7-s + 1.79·8-s + 0.475·10-s − 0.0413·11-s − 13-s + 1.05·14-s + 2.69·16-s − 3.33·17-s + 4.12·19-s + 1.77·20-s + 0.0196·22-s + 3.36·23-s + 25-s + 0.475·26-s + 3.92·28-s + 0.807·29-s − 6.11·31-s − 4.87·32-s + 1.58·34-s + 2.21·35-s − 3.50·37-s − 1.96·38-s − 1.79·40-s − 8.06·41-s + ⋯ |
| L(s) = 1 | − 0.336·2-s − 0.886·4-s − 0.447·5-s − 0.835·7-s + 0.635·8-s + 0.150·10-s − 0.0124·11-s − 0.277·13-s + 0.281·14-s + 0.672·16-s − 0.808·17-s + 0.946·19-s + 0.396·20-s + 0.00419·22-s + 0.701·23-s + 0.200·25-s + 0.0933·26-s + 0.741·28-s + 0.149·29-s − 1.09·31-s − 0.861·32-s + 0.272·34-s + 0.373·35-s − 0.576·37-s − 0.318·38-s − 0.283·40-s − 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5834426541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5834426541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.475T + 2T^{2} \) |
| 7 | \( 1 + 2.21T + 7T^{2} \) |
| 11 | \( 1 + 0.0413T + 11T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 - 0.807T + 29T^{2} \) |
| 31 | \( 1 + 6.11T + 31T^{2} \) |
| 37 | \( 1 + 3.50T + 37T^{2} \) |
| 41 | \( 1 + 8.06T + 41T^{2} \) |
| 43 | \( 1 - 1.54T + 43T^{2} \) |
| 47 | \( 1 + 1.29T + 47T^{2} \) |
| 53 | \( 1 + 5.00T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 7.44T + 61T^{2} \) |
| 67 | \( 1 + 6.95T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 8.22T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 7.75T + 83T^{2} \) |
| 89 | \( 1 - 5.56T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.260652045572901822077161308325, −7.51959883136492451303175983717, −6.93764903882601252366095828291, −6.07353391182433453736572961430, −5.11064154836966958036056877514, −4.61303393956228556670281575757, −3.59020776153190738326444582399, −3.10270654359462162666575256545, −1.69857668308643484450810244831, −0.43793449324495058005503781656,
0.43793449324495058005503781656, 1.69857668308643484450810244831, 3.10270654359462162666575256545, 3.59020776153190738326444582399, 4.61303393956228556670281575757, 5.11064154836966958036056877514, 6.07353391182433453736572961430, 6.93764903882601252366095828291, 7.51959883136492451303175983717, 8.260652045572901822077161308325
