| L(s) = 1 | − 2-s + 2.66·3-s + 4-s + 1.70·5-s − 2.66·6-s − 0.855·7-s − 8-s + 4.10·9-s − 1.70·10-s + 5.83·11-s + 2.66·12-s + 3.52·13-s + 0.855·14-s + 4.53·15-s + 16-s − 5.68·17-s − 4.10·18-s + 3.55·19-s + 1.70·20-s − 2.28·21-s − 5.83·22-s + 23-s − 2.66·24-s − 2.10·25-s − 3.52·26-s + 2.95·27-s − 0.855·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.53·3-s + 0.5·4-s + 0.760·5-s − 1.08·6-s − 0.323·7-s − 0.353·8-s + 1.36·9-s − 0.537·10-s + 1.75·11-s + 0.769·12-s + 0.976·13-s + 0.228·14-s + 1.17·15-s + 0.250·16-s − 1.37·17-s − 0.968·18-s + 0.816·19-s + 0.380·20-s − 0.497·21-s − 1.24·22-s + 0.208·23-s − 0.544·24-s − 0.421·25-s − 0.690·26-s + 0.568·27-s − 0.161·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.533928587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.533928587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 + 0.855T + 7T^{2} \) |
| 11 | \( 1 - 5.83T + 11T^{2} \) |
| 13 | \( 1 - 3.52T + 13T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 31 | \( 1 + 3.83T + 31T^{2} \) |
| 37 | \( 1 + 9.92T + 37T^{2} \) |
| 41 | \( 1 + 6.21T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 - 6.34T + 47T^{2} \) |
| 53 | \( 1 - 4.81T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 0.621T + 67T^{2} \) |
| 71 | \( 1 + 6.73T + 71T^{2} \) |
| 73 | \( 1 - 3.71T + 73T^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 8.61T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−9.412822365287383699746502802371, −8.813694155251841221080475240223, −8.474346423564922691273461179248, −7.16900749068537145829235241399, −6.69367746447867301357175784922, −5.68460819453052465664513667257, −4.03103977610790838614928835530, −3.39808976779849376405891180813, −2.19139929388222084382672273998, −1.42519868465509045056735806221,
1.42519868465509045056735806221, 2.19139929388222084382672273998, 3.39808976779849376405891180813, 4.03103977610790838614928835530, 5.68460819453052465664513667257, 6.69367746447867301357175784922, 7.16900749068537145829235241399, 8.474346423564922691273461179248, 8.813694155251841221080475240223, 9.412822365287383699746502802371
