| L(s) = 1 | + (−0.618 + 1.90i)2-s + (−6.12 + 4.45i)3-s + (−3.23 − 2.35i)4-s + (1.67 + 5.14i)5-s + (−4.68 − 14.4i)6-s + (17.9 + 13.0i)7-s + (6.47 − 4.70i)8-s + (9.39 − 28.9i)9-s − 10.8·10-s + (−11.0 + 34.7i)11-s + 30.3·12-s + (23.7 − 73.0i)13-s + (−35.8 + 26.0i)14-s + (−33.1 − 24.0i)15-s + (4.94 + 15.2i)16-s + (18.3 + 56.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−1.17 + 0.856i)3-s + (−0.404 − 0.293i)4-s + (0.149 + 0.460i)5-s + (−0.318 − 0.980i)6-s + (0.967 + 0.702i)7-s + (0.286 − 0.207i)8-s + (0.347 − 1.07i)9-s − 0.342·10-s + (−0.302 + 0.953i)11-s + 0.728·12-s + (0.506 − 1.55i)13-s + (−0.684 + 0.497i)14-s + (−0.570 − 0.414i)15-s + (0.0772 + 0.237i)16-s + (0.261 + 0.804i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.313334 + 0.656778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.313334 + 0.656778i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.618 - 1.90i)T \) |
| 11 | \( 1 + (11.0 - 34.7i)T \) |
| good | 3 | \( 1 + (6.12 - 4.45i)T + (8.34 - 25.6i)T^{2} \) |
| 5 | \( 1 + (-1.67 - 5.14i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (-17.9 - 13.0i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (-23.7 + 73.0i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-18.3 - 56.3i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (77.0 - 55.9i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-16.5 - 12.0i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-65.9 + 202. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-117. - 85.5i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (67.0 - 48.6i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (73.1 - 53.1i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (72.5 - 223. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (244. + 177. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (46.1 + 141. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 - 826.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (277. + 854. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-111. - 81.1i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (94.0 - 289. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (236. + 726. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 313.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (180. - 553. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
| 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
−17.67863521607243215670994087111, −16.89157980752764458458304136574, −15.30330045779383103834161107708, −14.95027215575722735522625753928, −12.66446691969587191836063966688, −11.01105744352734982325271591965, −10.12867474646955865261162915617, −8.160973284074851906535174974462, −6.06402805807492753657783912086, −4.87669696456338650979064339367,
1.11978324879953669972392833125, 4.91856253585733280620776097713, 6.91257774629780427196959960341, 8.749319767119791034890050419680, 10.96982672320137755576334432116, 11.50674989118800180014276782730, 12.95013434486640666311543577358, 14.00638485873588345696455320639, 16.52517060731106987808948979171, 17.20317249667413896873862529866
