| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s − 2·7-s − 2·9-s − 2·10-s − 11-s + 2·12-s − 4·13-s + 4·14-s + 15-s − 4·16-s + 2·17-s + 4·18-s + 2·20-s − 2·21-s + 2·22-s + 23-s − 4·25-s + 8·26-s − 5·27-s − 4·28-s − 2·30-s + 8·32-s − 33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s − 0.755·7-s − 2/3·9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s − 1.10·13-s + 1.06·14-s + 0.258·15-s − 16-s + 0.485·17-s + 0.942·18-s + 0.447·20-s − 0.436·21-s + 0.426·22-s + 0.208·23-s − 4/5·25-s + 1.56·26-s − 0.962·27-s − 0.755·28-s − 0.365·30-s + 1.41·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10571 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10571 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5240223981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5240223981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−16.77449892226152, −16.26568037016838, −15.40256211091329, −15.09978743688604, −14.16282539303016, −13.79888603676741, −13.23153109292678, −12.41430074030759, −11.86440909759066, −11.12991247252332, −10.32546209246895, −10.03274299392999, −9.411059901747954, −8.991115045082956, −8.402195617312077, −7.642530099395987, −7.326732520987610, −6.471312842444767, −5.728539969028606, −5.013138275158901, −3.969820132113129, −3.005868135704493, −2.442382402568148, −1.658447477022051, −0.3946435216621334,
0.3946435216621334, 1.658447477022051, 2.442382402568148, 3.005868135704493, 3.969820132113129, 5.013138275158901, 5.728539969028606, 6.471312842444767, 7.326732520987610, 7.642530099395987, 8.402195617312077, 8.991115045082956, 9.411059901747954, 10.03274299392999, 10.32546209246895, 11.12991247252332, 11.86440909759066, 12.41430074030759, 13.23153109292678, 13.79888603676741, 14.16282539303016, 15.09978743688604, 15.40256211091329, 16.26568037016838, 16.77449892226152
