| L(s) = 1 | + 4.50e4·3-s + 9.76e6·5-s − 7.62e8·7-s − 8.43e9·9-s + 5.05e10·11-s − 5.84e11·13-s + 4.39e11·15-s + 1.51e13·17-s − 1.45e12·19-s − 3.43e13·21-s − 2.85e14·23-s + 9.53e13·25-s − 8.50e14·27-s − 2.16e15·29-s + 4.84e15·31-s + 2.27e15·33-s − 7.44e15·35-s − 1.27e16·37-s − 2.63e16·39-s − 9.07e16·41-s − 1.68e17·43-s − 8.23e16·45-s + 6.66e17·47-s + 2.27e16·49-s + 6.80e17·51-s + 1.01e18·53-s + 4.93e17·55-s + ⋯ |
| L(s) = 1 | + 0.440·3-s + 0.447·5-s − 1.02·7-s − 0.806·9-s + 0.587·11-s − 1.17·13-s + 0.196·15-s + 1.81·17-s − 0.0545·19-s − 0.449·21-s − 1.43·23-s + 0.199·25-s − 0.795·27-s − 0.956·29-s + 1.06·31-s + 0.258·33-s − 0.456·35-s − 0.435·37-s − 0.518·39-s − 1.05·41-s − 1.18·43-s − 0.360·45-s + 1.84·47-s + 0.0407·49-s + 0.800·51-s + 0.793·53-s + 0.262·55-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
| \(L(11)\) |
\(\approx\) |
\(1.772877574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.772877574\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 4.50e4T + 1.04e10T^{2} \) |
| 7 | \( 1 + 7.62e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 5.05e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 5.84e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.51e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.45e12T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.85e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.16e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 4.84e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.27e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 9.07e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.68e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 6.66e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.01e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 2.57e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 6.48e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.88e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.24e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 5.64e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 2.38e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.16e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.23e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.24e20T + 5.27e41T^{2} \) |
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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
−10.07691026926515609779361500054, −9.673798050738312680552209340879, −8.487131272171996628119452946679, −7.37466704246149978674767543670, −6.18729549529693712964522610994, −5.33938525777415631899233953864, −3.75171853311346750008231722498, −2.93127143920594110063648703756, −1.91841928842008465558939136957, −0.51071961930440963974894538350,
0.51071961930440963974894538350, 1.91841928842008465558939136957, 2.93127143920594110063648703756, 3.75171853311346750008231722498, 5.33938525777415631899233953864, 6.18729549529693712964522610994, 7.37466704246149978674767543670, 8.487131272171996628119452946679, 9.673798050738312680552209340879, 10.07691026926515609779361500054
