| L(s) = 1 | + 0.582·3-s − 3.85·5-s − 7-s − 2.66·9-s − 4.64·11-s − 2.54·13-s − 2.24·15-s − 17-s − 6.76·19-s − 0.582·21-s + 6.10·23-s + 9.83·25-s − 3.29·27-s + 3.73·29-s − 0.0519·31-s − 2.70·33-s + 3.85·35-s − 10.1·37-s − 1.48·39-s − 5.63·41-s − 8.99·43-s + 10.2·45-s − 6.70·47-s + 49-s − 0.582·51-s + 8.65·53-s + 17.8·55-s + ⋯ |
| L(s) = 1 | + 0.336·3-s − 1.72·5-s − 0.377·7-s − 0.886·9-s − 1.40·11-s − 0.706·13-s − 0.579·15-s − 0.242·17-s − 1.55·19-s − 0.127·21-s + 1.27·23-s + 1.96·25-s − 0.634·27-s + 0.694·29-s − 0.00932·31-s − 0.471·33-s + 0.650·35-s − 1.67·37-s − 0.237·39-s − 0.880·41-s − 1.37·43-s + 1.52·45-s − 0.978·47-s + 0.142·49-s − 0.0815·51-s + 1.18·53-s + 2.41·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3094178409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3094178409\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.582T + 3T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 19 | \( 1 + 6.76T + 19T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 + 0.0519T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 5.63T + 41T^{2} \) |
| 43 | \( 1 + 8.99T + 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 - 8.65T + 53T^{2} \) |
| 59 | \( 1 + 0.380T + 59T^{2} \) |
| 61 | \( 1 - 4.49T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 + 4.35T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 - 9.48T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−8.504708590128670577854074880859, −7.898654341371264773762748271848, −7.11702342398126107518013229415, −6.53846479727663013626246762803, −5.19655664967236381065948114237, −4.79285062487045666522811837181, −3.68517672044222529327403898490, −3.08809053841196655082215872324, −2.29092901338734130014451993169, −0.29111587629453793691566558876,
0.29111587629453793691566558876, 2.29092901338734130014451993169, 3.08809053841196655082215872324, 3.68517672044222529327403898490, 4.79285062487045666522811837181, 5.19655664967236381065948114237, 6.53846479727663013626246762803, 7.11702342398126107518013229415, 7.898654341371264773762748271848, 8.504708590128670577854074880859
