| L(s) = 1 | − 0.976·2-s + 0.347·3-s − 1.04·4-s − 2.45·5-s − 0.339·6-s − 4.72·7-s + 2.97·8-s − 2.87·9-s + 2.40·10-s + 11-s − 0.363·12-s − 3.64·13-s + 4.61·14-s − 0.854·15-s − 0.815·16-s + 7.55·17-s + 2.81·18-s + 3.11·19-s + 2.57·20-s − 1.63·21-s − 0.976·22-s + 6.22·23-s + 1.03·24-s + 1.05·25-s + 3.55·26-s − 2.04·27-s + 4.93·28-s + ⋯ |
| L(s) = 1 | − 0.690·2-s + 0.200·3-s − 0.522·4-s − 1.10·5-s − 0.138·6-s − 1.78·7-s + 1.05·8-s − 0.959·9-s + 0.759·10-s + 0.301·11-s − 0.104·12-s − 1.01·13-s + 1.23·14-s − 0.220·15-s − 0.203·16-s + 1.83·17-s + 0.663·18-s + 0.715·19-s + 0.575·20-s − 0.357·21-s − 0.208·22-s + 1.29·23-s + 0.210·24-s + 0.210·25-s + 0.698·26-s − 0.392·27-s + 0.933·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4202290674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4202290674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.976T + 2T^{2} \) |
| 3 | \( 1 - 0.347T + 3T^{2} \) |
| 5 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 + 4.72T + 7T^{2} \) |
| 13 | \( 1 + 3.64T + 13T^{2} \) |
| 17 | \( 1 - 7.55T + 17T^{2} \) |
| 19 | \( 1 - 3.11T + 19T^{2} \) |
| 23 | \( 1 - 6.22T + 23T^{2} \) |
| 29 | \( 1 + 9.00T + 29T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 - 1.37T + 37T^{2} \) |
| 41 | \( 1 - 7.01T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 - 0.462T + 47T^{2} \) |
| 53 | \( 1 + 8.75T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 67 | \( 1 + 2.33T + 67T^{2} \) |
| 71 | \( 1 + 3.19T + 71T^{2} \) |
| 73 | \( 1 - 6.46T + 73T^{2} \) |
| 79 | \( 1 - 7.48T + 79T^{2} \) |
| 83 | \( 1 - 7.42T + 83T^{2} \) |
| 89 | \( 1 - 2.62T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−10.17889197686199067877760305880, −9.472794331728089839062303703144, −9.036293797621541320250354999890, −7.72390992683616466850058869860, −7.50624916619361901427342453826, −6.11668430782022597139033265307, −4.99653949861648251892627335841, −3.62469029781658922255967904342, −3.05674255252205889455810454842, −0.58698112200154550258595656052,
0.58698112200154550258595656052, 3.05674255252205889455810454842, 3.62469029781658922255967904342, 4.99653949861648251892627335841, 6.11668430782022597139033265307, 7.50624916619361901427342453826, 7.72390992683616466850058869860, 9.036293797621541320250354999890, 9.472794331728089839062303703144, 10.17889197686199067877760305880
