| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 4·7-s − 8-s + 9-s + 2·10-s − 11-s − 12-s + 4·14-s + 2·15-s + 16-s + 17-s − 18-s − 4·19-s − 2·20-s + 4·21-s + 22-s − 4·23-s + 24-s − 25-s − 27-s − 4·28-s − 6·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s + 1.06·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.872·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.54514147880518, −12.96581817066655, −12.45639637640687, −12.06780360122653, −11.86735447728206, −11.08933287313832, −10.75945028641256, −10.22575236357951, −9.892714230932167, −9.396637863681827, −8.867804720261815, −8.303908106665658, −7.878684171209279, −7.302435478492502, −6.881186828185882, −6.466858833252671, −5.820662145964507, −5.591270092191746, −4.688621184882303, −3.988792267692107, −3.739802495533634, −3.060738624222584, −2.434693391121293, −1.774165614752060, −0.8537392533016001, 0, 0,
0.8537392533016001, 1.774165614752060, 2.434693391121293, 3.060738624222584, 3.739802495533634, 3.988792267692107, 4.688621184882303, 5.591270092191746, 5.820662145964507, 6.466858833252671, 6.881186828185882, 7.302435478492502, 7.878684171209279, 8.303908106665658, 8.867804720261815, 9.396637863681827, 9.892714230932167, 10.22575236357951, 10.75945028641256, 11.08933287313832, 11.86735447728206, 12.06780360122653, 12.45639637640687, 12.96581817066655, 13.54514147880518
