| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s + 7-s − 8-s + 9-s − 3·10-s − 11-s + 12-s + 2·13-s − 14-s + 3·15-s + 16-s − 18-s + 4·19-s + 3·20-s + 21-s + 22-s − 5·23-s − 24-s + 4·25-s − 2·26-s + 27-s + 28-s − 10·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.670·20-s + 0.218·21-s + 0.213·22-s − 1.04·23-s − 0.204·24-s + 4/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 4091 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.79326564669052, −15.06091591761003, −14.59313116316053, −14.01081178633398, −13.57891800624900, −13.11509509900208, −12.54394030816877, −11.64589651518677, −11.34220514056081, −10.44783352509568, −10.15348384375463, −9.564207391219277, −9.147674290447862, −8.572789060827032, −7.913575537906088, −7.496457247614643, −6.679933066769394, −6.158609517516783, −5.433291539356207, −5.054890942462339, −3.881318458411582, −3.298031810255898, −2.452946212143780, −1.712285365003601, −1.464303887623937, 0,
1.464303887623937, 1.712285365003601, 2.452946212143780, 3.298031810255898, 3.881318458411582, 5.054890942462339, 5.433291539356207, 6.158609517516783, 6.679933066769394, 7.496457247614643, 7.913575537906088, 8.572789060827032, 9.147674290447862, 9.564207391219277, 10.15348384375463, 10.44783352509568, 11.34220514056081, 11.64589651518677, 12.54394030816877, 13.11509509900208, 13.57891800624900, 14.01081178633398, 14.59313116316053, 15.06091591761003, 15.79326564669052
