| L(s) = 1 | − 0.618·2-s + 2.23·3-s − 1.61·4-s + 3.23·5-s − 1.38·6-s − 3·7-s + 2.23·8-s + 2.00·9-s − 2.00·10-s + 5·11-s − 3.61·12-s + 2.38·13-s + 1.85·14-s + 7.23·15-s + 1.85·16-s − 2.23·17-s − 1.23·18-s − 0.763·19-s − 5.23·20-s − 6.70·21-s − 3.09·22-s + 0.618·23-s + 5.00·24-s + 5.47·25-s − 1.47·26-s − 2.23·27-s + 4.85·28-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 1.29·3-s − 0.809·4-s + 1.44·5-s − 0.564·6-s − 1.13·7-s + 0.790·8-s + 0.666·9-s − 0.632·10-s + 1.50·11-s − 1.04·12-s + 0.660·13-s + 0.495·14-s + 1.86·15-s + 0.463·16-s − 0.542·17-s − 0.291·18-s − 0.175·19-s − 1.17·20-s − 1.46·21-s − 0.658·22-s + 0.128·23-s + 1.02·24-s + 1.09·25-s − 0.288·26-s − 0.430·27-s + 0.917·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.084239716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084239716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1063 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 + 0.763T + 19T^{2} \) |
| 23 | \( 1 - 0.618T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 - 3.76T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 + 0.854T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 + 8.47T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 + 6.38T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 1.38T + 79T^{2} \) |
| 83 | \( 1 + 9.61T + 83T^{2} \) |
| 89 | \( 1 + 4.09T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−9.596434007223015287469665801236, −9.068508963043712439780586280530, −8.774301188145553377084404047024, −7.61027814053336354935909877855, −6.44126689014444611770879975041, −5.91089913544347253502704873600, −4.38102725589129454490154931458, −3.54742477971711743768638551622, −2.45789966962892763035910797187, −1.26184393055945050963951871930,
1.26184393055945050963951871930, 2.45789966962892763035910797187, 3.54742477971711743768638551622, 4.38102725589129454490154931458, 5.91089913544347253502704873600, 6.44126689014444611770879975041, 7.61027814053336354935909877855, 8.774301188145553377084404047024, 9.068508963043712439780586280530, 9.596434007223015287469665801236
