L(s) = 1 | + 2-s + 2.79·3-s − 4-s + 1.79·5-s + 2.79·6-s + 7-s − 3·8-s + 4.79·9-s + 1.79·10-s − 11-s − 2.79·12-s − 13-s + 14-s + 5·15-s − 16-s + 6.79·17-s + 4.79·18-s + 0.208·19-s − 1.79·20-s + 2.79·21-s − 22-s + 1.58·23-s − 8.37·24-s − 1.79·25-s − 26-s + 4.99·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.61·3-s − 0.5·4-s + 0.801·5-s + 1.13·6-s + 0.377·7-s − 1.06·8-s + 1.59·9-s + 0.566·10-s − 0.301·11-s − 0.805·12-s − 0.277·13-s + 0.267·14-s + 1.29·15-s − 0.250·16-s + 1.64·17-s + 1.12·18-s + 0.0478·19-s − 0.400·20-s + 0.609·21-s − 0.213·22-s + 0.329·23-s − 1.70·24-s − 0.358·25-s − 0.196·26-s + 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.808486026\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.808486026\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 - 1.79T + 5T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 - 0.208T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 7.16T + 37T^{2} \) |
| 41 | \( 1 + 7.16T + 41T^{2} \) |
| 43 | \( 1 + 2.79T + 43T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 + 5.20T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 0.373T + 67T^{2} \) |
| 71 | \( 1 - 4.20T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 7.79T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 5.20T + 89T^{2} \) |
| 97 | \( 1 + 5.16T + 97T^{2} \) |
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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
−9.785248861637203306037989657248, −9.168817273615209587172513025739, −8.297078141937075251300385761919, −7.78145727609793546163302749259, −6.50938497973591944108180247995, −5.39534321349643862087484800735, −4.66675548399475260983610422737, −3.45150607629041504449767381803, −2.88737154148015394605782579257, −1.61771602196479052725853611656,
1.61771602196479052725853611656, 2.88737154148015394605782579257, 3.45150607629041504449767381803, 4.66675548399475260983610422737, 5.39534321349643862087484800735, 6.50938497973591944108180247995, 7.78145727609793546163302749259, 8.297078141937075251300385761919, 9.168817273615209587172513025739, 9.785248861637203306037989657248