| L(s) = 1 | + 1.07·3-s − 3.37·5-s + 5.03·7-s − 1.84·9-s − 0.757·11-s − 5.91·13-s − 3.63·15-s − 0.402·17-s − 1.31·19-s + 5.42·21-s + 4.16·23-s + 6.37·25-s − 5.21·27-s + 4.50·29-s − 4.73·31-s − 0.815·33-s − 16.9·35-s + 0.609·37-s − 6.36·39-s + 2.90·41-s − 3.27·43-s + 6.20·45-s − 7.08·47-s + 18.3·49-s − 0.433·51-s + 10.6·53-s + 2.55·55-s + ⋯ |
| L(s) = 1 | + 0.621·3-s − 1.50·5-s + 1.90·7-s − 0.613·9-s − 0.228·11-s − 1.64·13-s − 0.937·15-s − 0.0975·17-s − 0.300·19-s + 1.18·21-s + 0.869·23-s + 1.27·25-s − 1.00·27-s + 0.836·29-s − 0.851·31-s − 0.141·33-s − 2.87·35-s + 0.100·37-s − 1.01·39-s + 0.454·41-s − 0.499·43-s + 0.925·45-s − 1.03·47-s + 2.62·49-s − 0.0606·51-s + 1.45·53-s + 0.344·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.669679218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669679218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.07T + 3T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 + 0.757T + 11T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 + 0.402T + 17T^{2} \) |
| 19 | \( 1 + 1.31T + 19T^{2} \) |
| 23 | \( 1 - 4.16T + 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 - 0.609T + 37T^{2} \) |
| 41 | \( 1 - 2.90T + 41T^{2} \) |
| 43 | \( 1 + 3.27T + 43T^{2} \) |
| 47 | \( 1 + 7.08T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 1.30T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 3.47T + 67T^{2} \) |
| 71 | \( 1 - 5.95T + 71T^{2} \) |
| 73 | \( 1 - 4.14T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 4.81T + 89T^{2} \) |
| 97 | \( 1 + 2.06T + 97T^{2} \) |
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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
−7.73757074284163726466264173581, −7.58138050995230343774954253123, −6.75925601869413423250113535867, −5.40348797359850201781401097107, −4.90144048646035203090443470718, −4.37916707031059734620022032239, −3.52238225304241818060170660948, −2.64991614876199849960284174821, −1.95016127353942274830314788455, −0.60261062852479180735688564665,
0.60261062852479180735688564665, 1.95016127353942274830314788455, 2.64991614876199849960284174821, 3.52238225304241818060170660948, 4.37916707031059734620022032239, 4.90144048646035203090443470718, 5.40348797359850201781401097107, 6.75925601869413423250113535867, 7.58138050995230343774954253123, 7.73757074284163726466264173581
