| L(s) = 1 | − 2·4-s − 6·11-s + 4·16-s + 6·17-s + 19-s + 6·23-s − 5·25-s − 6·29-s − 5·31-s + 37-s − 4·43-s + 12·44-s + 6·47-s − 6·53-s − 13·61-s − 8·64-s + 13·67-s − 12·68-s − 12·71-s + 10·73-s − 2·76-s − 79-s − 6·83-s + 6·89-s − 12·92-s − 5·97-s + 10·100-s + ⋯ |
| L(s) = 1 | − 4-s − 1.80·11-s + 16-s + 1.45·17-s + 0.229·19-s + 1.25·23-s − 25-s − 1.11·29-s − 0.898·31-s + 0.164·37-s − 0.609·43-s + 1.80·44-s + 0.875·47-s − 0.824·53-s − 1.66·61-s − 64-s + 1.58·67-s − 1.45·68-s − 1.42·71-s + 1.17·73-s − 0.229·76-s − 0.112·79-s − 0.658·83-s + 0.635·89-s − 1.25·92-s − 0.507·97-s + 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7832275586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7832275586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.86057263350306, −13.66199325974100, −13.00638743114826, −12.72515173977240, −12.28955206006818, −11.58053855791361, −10.88001892475863, −10.60219177933166, −9.947632396899927, −9.511075943420923, −9.167789347824530, −8.293516771377320, −8.032833138987484, −7.484225432455785, −7.120860466158450, −6.004020082987657, −5.542956296398518, −5.265189083033297, −4.697563621217934, −3.953267079569401, −3.305837382071309, −2.926723319168016, −1.997470694185423, −1.196593930838124, −0.3178919563286053,
0.3178919563286053, 1.196593930838124, 1.997470694185423, 2.926723319168016, 3.305837382071309, 3.953267079569401, 4.697563621217934, 5.265189083033297, 5.542956296398518, 6.004020082987657, 7.120860466158450, 7.484225432455785, 8.032833138987484, 8.293516771377320, 9.167789347824530, 9.511075943420923, 9.947632396899927, 10.60219177933166, 10.88001892475863, 11.58053855791361, 12.28955206006818, 12.72515173977240, 13.00638743114826, 13.66199325974100, 13.86057263350306
