| L(s) = 1 | − 2.17·2-s + 2.72·4-s − 2.40·5-s + 4.30·7-s − 1.57·8-s + 5.22·10-s + 4.13·13-s − 9.35·14-s − 2.02·16-s − 2.34·17-s + 6.03·19-s − 6.55·20-s + 4.18·23-s + 0.783·25-s − 8.99·26-s + 11.7·28-s − 8.53·29-s − 3.24·31-s + 7.54·32-s + 5.09·34-s − 10.3·35-s − 10.8·37-s − 13.1·38-s + 3.79·40-s − 3.39·41-s + 5.86·43-s − 9.08·46-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.36·4-s − 1.07·5-s + 1.62·7-s − 0.558·8-s + 1.65·10-s + 1.14·13-s − 2.50·14-s − 0.505·16-s − 0.567·17-s + 1.38·19-s − 1.46·20-s + 0.871·23-s + 0.156·25-s − 1.76·26-s + 2.21·28-s − 1.58·29-s − 0.583·31-s + 1.33·32-s + 0.873·34-s − 1.74·35-s − 1.77·37-s − 2.12·38-s + 0.600·40-s − 0.530·41-s + 0.894·43-s − 1.33·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9449559015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9449559015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 - 4.30T + 7T^{2} \) |
| 13 | \( 1 - 4.13T + 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 19 | \( 1 - 6.03T + 19T^{2} \) |
| 23 | \( 1 - 4.18T + 23T^{2} \) |
| 29 | \( 1 + 8.53T + 29T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 3.39T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 - 5.76T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 4.81T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 0.128T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 0.375T + 97T^{2} \) |
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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
−7.72551984693040624148601103577, −7.40708108380242858936490041721, −6.78971612428197162835122036083, −5.57924172058740117056291204058, −4.99628736929187293415891494797, −4.06277709148928576692927802689, −3.44749116562095788735826394133, −2.12931686193637134895596809659, −1.44479616380178818307670248672, −0.64172068790879128690482182511,
0.64172068790879128690482182511, 1.44479616380178818307670248672, 2.12931686193637134895596809659, 3.44749116562095788735826394133, 4.06277709148928576692927802689, 4.99628736929187293415891494797, 5.57924172058740117056291204058, 6.78971612428197162835122036083, 7.40708108380242858936490041721, 7.72551984693040624148601103577
