| L(s) = 1 | − 42.6·3-s − 1.23e3·5-s − 892.·7-s − 1.78e4·9-s + 2.71e4·11-s − 2.85e4·13-s + 5.26e4·15-s − 3.46e4·17-s − 4.28e5·19-s + 3.80e4·21-s − 2.03e6·23-s − 4.24e5·25-s + 1.60e6·27-s − 5.26e6·29-s + 4.15e6·31-s − 1.15e6·33-s + 1.10e6·35-s − 7.58e6·37-s + 1.21e6·39-s − 4.92e6·41-s − 1.71e7·43-s + 2.20e7·45-s + 2.95e7·47-s − 3.95e7·49-s + 1.47e6·51-s − 2.72e7·53-s − 3.35e7·55-s + ⋯ |
| L(s) = 1 | − 0.303·3-s − 0.884·5-s − 0.140·7-s − 0.907·9-s + 0.559·11-s − 0.277·13-s + 0.268·15-s − 0.100·17-s − 0.755·19-s + 0.0426·21-s − 1.51·23-s − 0.217·25-s + 0.579·27-s − 1.38·29-s + 0.808·31-s − 0.169·33-s + 0.124·35-s − 0.665·37-s + 0.0842·39-s − 0.272·41-s − 0.766·43-s + 0.802·45-s + 0.883·47-s − 0.980·49-s + 0.0305·51-s − 0.474·53-s − 0.494·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5272077120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5272077120\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 3 | \( 1 + 42.6T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.23e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 892.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.71e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.46e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.28e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.03e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.26e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.15e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 7.58e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.92e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.71e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.95e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.72e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.13e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.76e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.90e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.87e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.61e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.42e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.80e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.59e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.46e9T + 7.60e17T^{2} \) |
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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
−10.94265848898050301665307464514, −9.793858629678726976310878331565, −8.632018342429978344092581271445, −7.84500977559538956793950979405, −6.63888985706275216207655922525, −5.66987305421206267286997996152, −4.35238939870551459522154408734, −3.41101567258497961492791786026, −1.99317597127186458009486422799, −0.33484440199865465095125333483,
0.33484440199865465095125333483, 1.99317597127186458009486422799, 3.41101567258497961492791786026, 4.35238939870551459522154408734, 5.66987305421206267286997996152, 6.63888985706275216207655922525, 7.84500977559538956793950979405, 8.632018342429978344092581271445, 9.793858629678726976310878331565, 10.94265848898050301665307464514
