| L(s) = 1 | + 0.564·3-s + 1.37·5-s + 0.329·7-s − 2.68·9-s − 2.14·13-s + 0.774·15-s + 2.03·17-s − 19-s + 0.185·21-s − 0.737·23-s − 3.11·25-s − 3.20·27-s + 7.92·29-s + 3.78·31-s + 0.452·35-s + 0.421·37-s − 1.20·39-s − 4.47·41-s − 7.32·43-s − 3.67·45-s − 5.65·47-s − 6.89·49-s + 1.15·51-s + 11.6·53-s − 0.564·57-s + 10.2·59-s − 14.5·61-s + ⋯ |
| L(s) = 1 | + 0.325·3-s + 0.613·5-s + 0.124·7-s − 0.893·9-s − 0.593·13-s + 0.200·15-s + 0.494·17-s − 0.229·19-s + 0.0405·21-s − 0.153·23-s − 0.623·25-s − 0.617·27-s + 1.47·29-s + 0.679·31-s + 0.0764·35-s + 0.0692·37-s − 0.193·39-s − 0.699·41-s − 1.11·43-s − 0.548·45-s − 0.824·47-s − 0.984·49-s + 0.161·51-s + 1.60·53-s − 0.0747·57-s + 1.32·59-s − 1.86·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.564T + 3T^{2} \) |
| 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 - 0.329T + 7T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 17 | \( 1 - 2.03T + 17T^{2} \) |
| 23 | \( 1 + 0.737T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 3.78T + 31T^{2} \) |
| 37 | \( 1 - 0.421T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 7.32T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 9.51T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 5.94T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.48899311731376583343917713171, −6.54459642259861327805871947217, −6.10142336831714520080705292714, −5.26130771598868376800386404157, −4.75359076084669694377706745538, −3.71686153319973343015832880074, −2.91683009372135575609406667836, −2.30719668433841760238144389191, −1.35799861380384137452084794463, 0,
1.35799861380384137452084794463, 2.30719668433841760238144389191, 2.91683009372135575609406667836, 3.71686153319973343015832880074, 4.75359076084669694377706745538, 5.26130771598868376800386404157, 6.10142336831714520080705292714, 6.54459642259861327805871947217, 7.48899311731376583343917713171
