| L(s) = 1 | − 2-s + 0.347·3-s + 4-s + 0.852·5-s − 0.347·6-s + 1.38·7-s − 8-s − 2.87·9-s − 0.852·10-s + 0.932·11-s + 0.347·12-s + 3.64·13-s − 1.38·14-s + 0.296·15-s + 16-s + 4.42·17-s + 2.87·18-s + 3.82·19-s + 0.852·20-s + 0.480·21-s − 0.932·22-s − 1.06·23-s − 0.347·24-s − 4.27·25-s − 3.64·26-s − 2.04·27-s + 1.38·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.200·3-s + 0.5·4-s + 0.381·5-s − 0.141·6-s + 0.523·7-s − 0.353·8-s − 0.959·9-s − 0.269·10-s + 0.281·11-s + 0.100·12-s + 1.01·13-s − 0.370·14-s + 0.0764·15-s + 0.250·16-s + 1.07·17-s + 0.678·18-s + 0.877·19-s + 0.190·20-s + 0.104·21-s − 0.198·22-s − 0.221·23-s − 0.0708·24-s − 0.854·25-s − 0.714·26-s − 0.392·27-s + 0.261·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.670966802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.670966802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 0.347T + 3T^{2} \) |
| 5 | \( 1 - 0.852T + 5T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 - 0.932T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 + 1.00T + 29T^{2} \) |
| 31 | \( 1 - 7.33T + 31T^{2} \) |
| 41 | \( 1 - 0.243T + 41T^{2} \) |
| 43 | \( 1 + 5.13T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 9.76T + 59T^{2} \) |
| 61 | \( 1 - 0.397T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 3.48T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 - 6.28T + 89T^{2} \) |
| 97 | \( 1 + 7.47T + 97T^{2} \) |
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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
−8.579563466722165942398602957968, −8.314748585661310427029472797326, −7.54845216432233714203854001212, −6.54198490845298049552300116116, −5.80963183283015820179398143224, −5.18500885584027825413041385194, −3.80996062064349031000753653424, −3.02531474541251843794735808554, −1.92857598976197342247850030590, −0.925480169397918778280261699795,
0.925480169397918778280261699795, 1.92857598976197342247850030590, 3.02531474541251843794735808554, 3.80996062064349031000753653424, 5.18500885584027825413041385194, 5.80963183283015820179398143224, 6.54198490845298049552300116116, 7.54845216432233714203854001212, 8.314748585661310427029472797326, 8.579563466722165942398602957968
