| L(s) = 1 | + 2.68·2-s + 5.18·4-s − 3.11·5-s − 2.37·7-s + 8.54·8-s − 8.33·10-s − 2.60·11-s − 2.30·13-s − 6.35·14-s + 12.5·16-s + 4.83·17-s + 1.63·19-s − 16.1·20-s − 6.97·22-s + 1.18·23-s + 4.67·25-s − 6.18·26-s − 12.2·28-s + 5.78·31-s + 16.5·32-s + 12.9·34-s + 7.37·35-s + 6.89·37-s + 4.39·38-s − 26.5·40-s + 7.66·41-s + 1.88·43-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 2.59·4-s − 1.39·5-s − 0.895·7-s + 3.02·8-s − 2.63·10-s − 0.784·11-s − 0.640·13-s − 1.69·14-s + 3.13·16-s + 1.17·17-s + 0.375·19-s − 3.60·20-s − 1.48·22-s + 0.246·23-s + 0.935·25-s − 1.21·26-s − 2.32·28-s + 1.03·31-s + 2.91·32-s + 2.22·34-s + 1.24·35-s + 1.13·37-s + 0.712·38-s − 4.20·40-s + 1.19·41-s + 0.287·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.883593383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.883593383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 5 | \( 1 + 3.11T + 5T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 19 | \( 1 - 1.63T + 19T^{2} \) |
| 23 | \( 1 - 1.18T + 23T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 - 6.89T + 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 - 1.88T + 43T^{2} \) |
| 47 | \( 1 - 9.30T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 - 4.90T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 6.97T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 - 3.75T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 5.70T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−7.56642209216144599213157669971, −7.18820415044976137831374794361, −6.24044363646530807898697391649, −5.71207410872747292624435922047, −4.87653677960708050048655603899, −4.33028237203314373418795855765, −3.58693147065194484515569604732, −3.02047267711475453741449472269, −2.44897044198483009649805511595, −0.808977519612790443892484021914,
0.808977519612790443892484021914, 2.44897044198483009649805511595, 3.02047267711475453741449472269, 3.58693147065194484515569604732, 4.33028237203314373418795855765, 4.87653677960708050048655603899, 5.71207410872747292624435922047, 6.24044363646530807898697391649, 7.18820415044976137831374794361, 7.56642209216144599213157669971
