| L(s) = 1 | + (−4.13 − 5.68i)3-s + (−10.7 − 3.12i)5-s + 21.6i·7-s + (−6.91 + 21.2i)9-s + (3.15 + 9.72i)11-s + (72.0 + 23.4i)13-s + (26.5 + 73.9i)15-s + (−77.2 + 106. i)17-s + (−93.9 − 68.2i)19-s + (123. − 89.6i)21-s + (−85.3 + 27.7i)23-s + (105. + 67.0i)25-s + (−30.8 + 10.0i)27-s + (−96.2 + 69.9i)29-s + (−153. − 111. i)31-s + ⋯ |
| L(s) = 1 | + (−0.794 − 1.09i)3-s + (−0.960 − 0.279i)5-s + 1.17i·7-s + (−0.256 + 0.788i)9-s + (0.0865 + 0.266i)11-s + (1.53 + 0.499i)13-s + (0.457 + 1.27i)15-s + (−1.10 + 1.51i)17-s + (−1.13 − 0.824i)19-s + (1.28 − 0.931i)21-s + (−0.774 + 0.251i)23-s + (0.843 + 0.536i)25-s + (−0.219 + 0.0714i)27-s + (−0.616 + 0.447i)29-s + (−0.889 − 0.645i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.256512 + 0.290708i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.256512 + 0.290708i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (10.7 + 3.12i)T \) |
| good | 3 | \( 1 + (4.13 + 5.68i)T + (-8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 - 21.6iT - 343T^{2} \) |
| 11 | \( 1 + (-3.15 - 9.72i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-72.0 - 23.4i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (77.2 - 106. i)T + (-1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (93.9 + 68.2i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (85.3 - 27.7i)T + (9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (96.2 - 69.9i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (153. + 111. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-207. - 67.5i)T + (4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (17.4 - 53.6i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 305. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (59.4 + 81.8i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (213. + 294. i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-57.9 + 178. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-48.2 - 148. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-264. + 364. i)T + (-9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (133. - 97.1i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (921. - 299. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (450. - 327. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (732. - 1.00e3i)T + (-1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (150. + 464. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-611. - 842. i)T + (-2.82e5 + 8.68e5i)T^{2} \) |
| 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
−13.01096502727749369994535467481, −12.78612391041416466770853920877, −11.49487800800581507911784401806, −11.13458581754351828257968367047, −8.948730068512732817584810631914, −8.161100704463914371558841877427, −6.67784059018053572080900690031, −5.89992342711323120648866341492, −4.12571826877823871887721587684, −1.77070094641744348126638512344,
0.24360820494761716660252829016, 3.72079387455838664987806040031, 4.43289919821243900346294090904, 6.06740382049234195332687448670, 7.42529584537575472141860130426, 8.767572272755626673636854463551, 10.31857014262053601262921497292, 10.91372777182820165728590550161, 11.57680548105931320702151555668, 13.11968756723022332365430491154
