| L(s) = 1 | − 3.10·3-s − 0.289·5-s + 4.81·7-s + 6.62·9-s + 5.62·11-s − 1.10·13-s + 0.897·15-s + 1.10·17-s + 6.44·19-s − 14.9·21-s + 7.10·23-s − 4.91·25-s − 11.2·27-s + 6.72·29-s − 31-s − 17.4·33-s − 1.39·35-s − 2·37-s + 3.42·39-s − 9.91·41-s − 0.318·43-s − 1.91·45-s − 9.45·47-s + 16.1·49-s − 3.42·51-s + 2.57·53-s − 1.62·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 0.129·5-s + 1.81·7-s + 2.20·9-s + 1.69·11-s − 0.305·13-s + 0.231·15-s + 0.267·17-s + 1.47·19-s − 3.25·21-s + 1.48·23-s − 0.983·25-s − 2.16·27-s + 1.24·29-s − 0.179·31-s − 3.03·33-s − 0.235·35-s − 0.328·37-s + 0.547·39-s − 1.54·41-s − 0.0486·43-s − 0.285·45-s − 1.37·47-s + 2.31·49-s − 0.479·51-s + 0.354·53-s − 0.219·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.428592847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.428592847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 + 0.289T + 5T^{2} \) |
| 7 | \( 1 - 4.81T + 7T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 - 1.10T + 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 - 7.10T + 23T^{2} \) |
| 29 | \( 1 - 6.72T + 29T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 9.91T + 41T^{2} \) |
| 43 | \( 1 + 0.318T + 43T^{2} \) |
| 47 | \( 1 + 9.45T + 47T^{2} \) |
| 53 | \( 1 - 2.57T + 53T^{2} \) |
| 59 | \( 1 + 8.06T + 59T^{2} \) |
| 61 | \( 1 - 3.30T + 61T^{2} \) |
| 67 | \( 1 + 9.62T + 67T^{2} \) |
| 71 | \( 1 - 5.97T + 71T^{2} \) |
| 73 | \( 1 - 0.843T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 5.62T + 83T^{2} \) |
| 89 | \( 1 + 1.68T + 89T^{2} \) |
| 97 | \( 1 + 8.75T + 97T^{2} \) |
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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
−9.296221769122409819932786492321, −8.288599786101390543944086697817, −7.35712538887309449571971717346, −6.78820158831074287452051306955, −5.88719464673001656462353114749, −4.93319156562683164353915769287, −4.79360571969800409442350764828, −3.60604786898247170877368058963, −1.59986416021830730997219217997, −1.01823672895751941671897208556,
1.01823672895751941671897208556, 1.59986416021830730997219217997, 3.60604786898247170877368058963, 4.79360571969800409442350764828, 4.93319156562683164353915769287, 5.88719464673001656462353114749, 6.78820158831074287452051306955, 7.35712538887309449571971717346, 8.288599786101390543944086697817, 9.296221769122409819932786492321
