| L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s − 2·13-s − 15-s − 2·17-s + 4·19-s + 25-s − 27-s + 2·29-s − 4·33-s + 10·37-s + 2·39-s − 10·41-s − 4·43-s + 45-s − 8·47-s + 2·51-s + 10·53-s + 4·55-s − 4·57-s − 4·59-s − 2·61-s − 2·65-s − 12·67-s − 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.696·33-s + 1.64·37-s + 0.320·39-s − 1.56·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 0.280·51-s + 1.37·53-s + 0.539·55-s − 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.248·65-s − 1.46·67-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.84197719724710, −14.41084788810987, −13.71656463097029, −13.34412453249119, −12.84991247527758, −12.06585615898560, −11.71868484052569, −11.48976823512617, −10.65279406353201, −10.09680027264216, −9.769360956976568, −9.061985381552094, −8.767742014465480, −7.866913446895062, −7.334022628481416, −6.749683045949990, −6.247677472354246, −5.827488630265298, −4.926591024246934, −4.712671845408465, −3.875132003461321, −3.220707643736083, −2.449861731927421, −1.605078511170438, −1.058671849247922, 0,
1.058671849247922, 1.605078511170438, 2.449861731927421, 3.220707643736083, 3.875132003461321, 4.712671845408465, 4.926591024246934, 5.827488630265298, 6.247677472354246, 6.749683045949990, 7.334022628481416, 7.866913446895062, 8.767742014465480, 9.061985381552094, 9.769360956976568, 10.09680027264216, 10.65279406353201, 11.48976823512617, 11.71868484052569, 12.06585615898560, 12.84991247527758, 13.34412453249119, 13.71656463097029, 14.41084788810987, 14.84197719724710
