| L(s) = 1 | − 3·9-s − 2·13-s − 4·19-s + 8·23-s − 5·25-s − 8·29-s − 8·31-s + 8·37-s + 8·41-s − 4·43-s − 8·47-s + 10·53-s + 12·59-s + 4·67-s − 8·71-s + 8·73-s + 8·79-s + 9·81-s + 4·83-s − 6·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·117-s + ⋯ |
| L(s) = 1 | − 9-s − 0.554·13-s − 0.917·19-s + 1.66·23-s − 25-s − 1.48·29-s − 1.43·31-s + 1.31·37-s + 1.24·41-s − 0.609·43-s − 1.16·47-s + 1.37·53-s + 1.56·59-s + 0.488·67-s − 0.949·71-s + 0.936·73-s + 0.900·79-s + 81-s + 0.439·83-s − 0.635·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.554·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.03265366122721, −12.97183316713470, −12.21019529989348, −11.68734109071944, −11.30962369985201, −10.89504853497858, −10.59085316361493, −9.714631339103396, −9.406695614601010, −9.093684417255729, −8.355015446609607, −8.141784892308781, −7.342767229093102, −7.170998119000047, −6.426581244046643, −5.958181519247066, −5.369416352443779, −5.159310104443254, −4.357558401686835, −3.799225205314417, −3.383382151916390, −2.510078191843739, −2.337110598067766, −1.539326852729474, −0.6530201556611124, 0,
0.6530201556611124, 1.539326852729474, 2.337110598067766, 2.510078191843739, 3.383382151916390, 3.799225205314417, 4.357558401686835, 5.159310104443254, 5.369416352443779, 5.958181519247066, 6.426581244046643, 7.170998119000047, 7.342767229093102, 8.141784892308781, 8.355015446609607, 9.093684417255729, 9.406695614601010, 9.714631339103396, 10.59085316361493, 10.89504853497858, 11.30962369985201, 11.68734109071944, 12.21019529989348, 12.97183316713470, 13.03265366122721
