| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 3·14-s − 15-s + 16-s − 17-s + 18-s + 4·19-s + 20-s − 3·21-s + 22-s − 23-s − 24-s + 25-s − 27-s + 3·28-s − 6·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.566·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 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
−14.04137534055935, −13.68390977132898, −13.31097629390170, −12.63856614997450, −12.11279685841046, −11.68368749514196, −11.36077267574277, −10.80789693874185, −10.37513520607811, −9.687048854044352, −9.319023413654416, −8.524961800884372, −7.966156591423220, −7.566144923747419, −6.828742979227241, −6.427267740335279, −5.867093834377500, −5.196687991252650, −4.926231147247056, −4.465993021935909, −3.597701421167805, −3.184050392727574, −2.195094966688803, −1.660605438843118, −1.187916383518170, 0,
1.187916383518170, 1.660605438843118, 2.195094966688803, 3.184050392727574, 3.597701421167805, 4.465993021935909, 4.926231147247056, 5.196687991252650, 5.867093834377500, 6.427267740335279, 6.828742979227241, 7.566144923747419, 7.966156591423220, 8.524961800884372, 9.319023413654416, 9.687048854044352, 10.37513520607811, 10.80789693874185, 11.36077267574277, 11.68368749514196, 12.11279685841046, 12.63856614997450, 13.31097629390170, 13.68390977132898, 14.04137534055935
