| L(s) = 1 | + 1.14e5·3-s − 9.76e6·5-s + 2.03e8·7-s + 2.72e9·9-s + 4.40e10·11-s − 1.80e11·13-s − 1.12e12·15-s + 1.29e13·17-s − 3.50e13·19-s + 2.34e13·21-s + 1.00e14·23-s + 9.53e13·25-s − 8.87e14·27-s + 2.65e15·29-s − 1.12e15·31-s + 5.05e15·33-s − 1.99e15·35-s + 2.62e16·37-s − 2.07e16·39-s − 2.80e16·41-s − 1.48e17·43-s − 2.66e16·45-s + 9.51e16·47-s − 5.16e17·49-s + 1.48e18·51-s + 1.03e18·53-s − 4.29e17·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s − 0.447·5-s + 0.272·7-s + 0.260·9-s + 0.511·11-s − 0.363·13-s − 0.502·15-s + 1.55·17-s − 1.31·19-s + 0.306·21-s + 0.505·23-s + 0.199·25-s − 0.829·27-s + 1.17·29-s − 0.246·31-s + 0.574·33-s − 0.122·35-s + 0.898·37-s − 0.408·39-s − 0.326·41-s − 1.04·43-s − 0.116·45-s + 0.263·47-s − 0.925·49-s + 1.74·51-s + 0.811·53-s − 0.228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(3.417528610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.417528610\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 9.76e6T \) |
| good | 3 | \( 1 - 1.14e5T + 1.04e10T^{2} \) |
| 7 | \( 1 - 2.03e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 4.40e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 1.80e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.29e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.50e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.00e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.65e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 1.12e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.62e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 2.80e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.48e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 9.51e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.03e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 1.03e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 8.22e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.74e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.67e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 5.60e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 4.33e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.59e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 5.09e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 6.58e20T + 5.27e41T^{2} \) |
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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
−10.38051331102989519490097030611, −9.285288101267410545192719918171, −8.346183674292668984613501733424, −7.67756037489016901414756718171, −6.41231007304462841593235643993, −4.95546869216508336769585692923, −3.80195553627326142863721600335, −2.96986903983968927230016128413, −1.90510772574495630555294512553, −0.71646660031784887901930839687,
0.71646660031784887901930839687, 1.90510772574495630555294512553, 2.96986903983968927230016128413, 3.80195553627326142863721600335, 4.95546869216508336769585692923, 6.41231007304462841593235643993, 7.67756037489016901414756718171, 8.346183674292668984613501733424, 9.285288101267410545192719918171, 10.38051331102989519490097030611
