| L(s) = 1 | − 1.73·2-s + 0.999·4-s − 3.46·5-s + 1.73·8-s + 5.99·10-s − 3.46·11-s − 2·13-s − 5·16-s + 3.46·17-s + 4·19-s − 3.46·20-s + 5.99·22-s + 3.46·23-s + 6.99·25-s + 3.46·26-s + 4·31-s + 5.19·32-s − 5.99·34-s + 2·37-s − 6.92·38-s − 6.00·40-s + 10.3·41-s − 4·43-s − 3.46·44-s − 5.99·46-s + 6.92·47-s − 12.1·50-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s − 1.54·5-s + 0.612·8-s + 1.89·10-s − 1.04·11-s − 0.554·13-s − 1.25·16-s + 0.840·17-s + 0.917·19-s − 0.774·20-s + 1.27·22-s + 0.722·23-s + 1.39·25-s + 0.679·26-s + 0.718·31-s + 0.918·32-s − 1.02·34-s + 0.328·37-s − 1.12·38-s − 0.948·40-s + 1.62·41-s − 0.609·43-s − 0.522·44-s − 0.884·46-s + 1.01·47-s − 1.71·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4408106826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4408106826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−10.95658054654902885523846772136, −10.17188386146879038957946866480, −9.268646300567467243411926807400, −8.225844865232840812413709801409, −7.69810934092326468321480626486, −7.11368080298575350370680898060, −5.29179036278425807274949680025, −4.23077073410533406981583562252, −2.85720623146223462659941831225, −0.74095505323005833418238937755,
0.74095505323005833418238937755, 2.85720623146223462659941831225, 4.23077073410533406981583562252, 5.29179036278425807274949680025, 7.11368080298575350370680898060, 7.69810934092326468321480626486, 8.225844865232840812413709801409, 9.268646300567467243411926807400, 10.17188386146879038957946866480, 10.95658054654902885523846772136
