L(s) = 1 | − 3.60i·2-s + 1.73i·3-s − 9.02·4-s + 2.97i·5-s + 6.25·6-s + 2.36i·7-s + 18.1i·8-s − 2.99·9-s + 10.7·10-s + 15.8i·11-s − 15.6i·12-s + 6.13i·13-s + 8.51·14-s − 5.14·15-s + 29.2·16-s + 11.1·17-s + ⋯ |
L(s) = 1 | − 1.80i·2-s + 0.577i·3-s − 2.25·4-s + 0.594i·5-s + 1.04·6-s + 0.337i·7-s + 2.26i·8-s − 0.333·9-s + 1.07·10-s + 1.44i·11-s − 1.30i·12-s + 0.471i·13-s + 0.608·14-s − 0.343·15-s + 1.83·16-s + 0.658·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05060 + 0.0392172i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05060 + 0.0392172i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 67 | \( 1 + (-4.99 - 66.8i)T \) |
good | 2 | \( 1 + 3.60iT - 4T^{2} \) |
| 5 | \( 1 - 2.97iT - 25T^{2} \) |
| 7 | \( 1 - 2.36iT - 49T^{2} \) |
| 11 | \( 1 - 15.8iT - 121T^{2} \) |
| 13 | \( 1 - 6.13iT - 169T^{2} \) |
| 17 | \( 1 - 11.1T + 289T^{2} \) |
| 19 | \( 1 - 18.6T + 361T^{2} \) |
| 23 | \( 1 + 24.0T + 529T^{2} \) |
| 29 | \( 1 + 48.8T + 841T^{2} \) |
| 31 | \( 1 - 34.1iT - 961T^{2} \) |
| 37 | \( 1 + 21.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 0.389iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 6.25iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 9.41T + 2.20e3T^{2} \) |
| 53 | \( 1 - 52.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 70.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 43.4iT - 3.72e3T^{2} \) |
| 71 | \( 1 - 8.27T + 5.04e3T^{2} \) |
| 73 | \( 1 - 28.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 107. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 151.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 99.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 164. iT - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.03440044328843391589219654827, −11.28438494479409472494632712937, −10.24468568825692842314703082426, −9.759918560614393520415783344282, −8.837563579884172843911052657216, −7.28469068457548950847399373469, −5.37413043003085715432765827456, −4.22214917725348375153680869316, −3.13173891748241004384590706640, −1.85806801482757117951887927512,
0.62873406673255338293836058152, 3.72101641336850145849886060343, 5.36575457557065269554920379871, 5.89424182825535122395812288439, 7.17735430260201995833604391744, 7.993067749516923517575149221255, 8.691933946230182880773359183525, 9.785475512725015144055630120578, 11.36038315744147098326268731366, 12.67594378982155702424651933546