| L(s) = 1 | − 163.·3-s + 1.92e3·5-s − 2.40e3·7-s + 7.02e3·9-s + 9.01e4·11-s − 3.19e3·13-s − 3.14e5·15-s + 1.16e5·17-s + 1.42e5·19-s + 3.92e5·21-s − 1.27e6·23-s + 1.74e6·25-s + 2.06e6·27-s − 1.42e6·29-s − 9.67e6·31-s − 1.47e7·33-s − 4.61e6·35-s − 8.67e6·37-s + 5.22e5·39-s + 1.32e7·41-s + 2.97e7·43-s + 1.34e7·45-s + 1.07e7·47-s + 5.76e6·49-s − 1.90e7·51-s + 7.07e7·53-s + 1.73e8·55-s + ⋯ |
| L(s) = 1 | − 1.16·3-s + 1.37·5-s − 0.377·7-s + 0.356·9-s + 1.85·11-s − 0.0310·13-s − 1.60·15-s + 0.338·17-s + 0.250·19-s + 0.440·21-s − 0.949·23-s + 0.891·25-s + 0.749·27-s − 0.375·29-s − 1.88·31-s − 2.16·33-s − 0.519·35-s − 0.761·37-s + 0.0361·39-s + 0.732·41-s + 1.32·43-s + 0.490·45-s + 0.322·47-s + 0.142·49-s − 0.394·51-s + 1.23·53-s + 2.55·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.833791895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.833791895\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 3 | \( 1 + 163.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.92e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 9.01e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.19e3T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.16e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.42e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.27e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.42e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.67e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.07e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.07e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.40e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.69e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.16e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.44e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.60e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.89e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 8.31e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.08e6T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.15e8T + 7.60e17T^{2} \) |
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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
−11.85008182115784568039152684903, −10.83083375820712642349634416672, −9.750036193513049679154958771447, −9.013026720279181921712496320317, −6.99614571838826159299367744698, −6.06837692564788067329713199691, −5.48042735980743809131722129479, −3.85048506004933971653121952462, −1.97238671442306773411372940468, −0.806361644195198712855562134940,
0.806361644195198712855562134940, 1.97238671442306773411372940468, 3.85048506004933971653121952462, 5.48042735980743809131722129479, 6.06837692564788067329713199691, 6.99614571838826159299367744698, 9.013026720279181921712496320317, 9.750036193513049679154958771447, 10.83083375820712642349634416672, 11.85008182115784568039152684903
