| L(s) = 1 | + 1.81·2-s + 0.177·3-s + 1.30·4-s + 0.321·6-s − 1.07·7-s − 1.26·8-s − 2.96·9-s − 3.31·11-s + 0.230·12-s + 2.65·13-s − 1.95·14-s − 4.90·16-s + 3.99·17-s − 5.39·18-s − 0.190·21-s − 6.01·22-s − 1.75·23-s − 0.224·24-s + 4.82·26-s − 1.05·27-s − 1.39·28-s − 2.25·29-s + 8.05·31-s − 6.38·32-s − 0.586·33-s + 7.26·34-s − 3.86·36-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 0.102·3-s + 0.650·4-s + 0.131·6-s − 0.405·7-s − 0.448·8-s − 0.989·9-s − 0.998·11-s + 0.0665·12-s + 0.737·13-s − 0.521·14-s − 1.22·16-s + 0.970·17-s − 1.27·18-s − 0.0414·21-s − 1.28·22-s − 0.365·23-s − 0.0458·24-s + 0.947·26-s − 0.203·27-s − 0.264·28-s − 0.418·29-s + 1.44·31-s − 1.12·32-s − 0.102·33-s + 1.24·34-s − 0.643·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.827144904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.827144904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.81T + 2T^{2} \) |
| 3 | \( 1 - 0.177T + 3T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 11 | \( 1 + 3.31T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 - 3.99T + 17T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 - 8.05T + 31T^{2} \) |
| 37 | \( 1 + 5.64T + 37T^{2} \) |
| 41 | \( 1 - 0.896T + 41T^{2} \) |
| 43 | \( 1 - 8.51T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 + 3.39T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 2.61T + 73T^{2} \) |
| 79 | \( 1 - 6.57T + 79T^{2} \) |
| 83 | \( 1 + 2.69T + 83T^{2} \) |
| 89 | \( 1 + 0.789T + 89T^{2} \) |
| 97 | \( 1 + 18.7T + 97T^{2} \) |
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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
−7.72210033224274346002280741524, −6.78457494275159569536617512259, −6.06243991240560992113573177760, −5.61036582888120176620867828032, −5.08694307438962019568523414837, −4.17623035453766930928179738807, −3.44977597530442987581169188281, −2.92782948371259525405090223167, −2.19844916312297285932190929755, −0.63205458132879530915859367975,
0.63205458132879530915859367975, 2.19844916312297285932190929755, 2.92782948371259525405090223167, 3.44977597530442987581169188281, 4.17623035453766930928179738807, 5.08694307438962019568523414837, 5.61036582888120176620867828032, 6.06243991240560992113573177760, 6.78457494275159569536617512259, 7.72210033224274346002280741524
