L(s) = 1 | − 2.04i·2-s + 3·3-s + 3.82·4-s − 12.0i·5-s − 6.12i·6-s + 29.7i·7-s − 24.1i·8-s + 9·9-s − 24.6·10-s + 28.0i·11-s + 11.4·12-s + 60.7·14-s − 36.2i·15-s − 18.7·16-s + 50.6·17-s − 18.3i·18-s + ⋯ |
L(s) = 1 | − 0.722i·2-s + 0.577·3-s + 0.478·4-s − 1.08i·5-s − 0.416i·6-s + 1.60i·7-s − 1.06i·8-s + 0.333·9-s − 0.780·10-s + 0.769i·11-s + 0.276·12-s + 1.15·14-s − 0.623i·15-s − 0.292·16-s + 0.722·17-s − 0.240i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.101661745\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.101661745\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.04iT - 8T^{2} \) |
| 5 | \( 1 + 12.0iT - 125T^{2} \) |
| 7 | \( 1 - 29.7iT - 343T^{2} \) |
| 11 | \( 1 - 28.0iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 50.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 105. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 223. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 228. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 295. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 192.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 36.9iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 438. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 286.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 537. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 102. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 75.5iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 17.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.46e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 334. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 748. iT - 9.12e5T^{2} \) |
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
−10.22267353550247595360647822279, −9.248978840970972990256445192678, −8.922892672075268648160115610428, −7.77329734925658518982573780592, −6.67842196620053186361295050558, −5.43707410833593134879105759008, −4.51128849840331514315772731375, −2.98577693412792605018405413666, −2.25599638478228663540414206937, −1.04139907702091437169516357153,
1.27465278832963619525818898568, 2.95533278924864039231553486448, 3.62897431713061192827330005912, 5.17816565800668850282710149616, 6.54550582310713542158572833971, 6.96506273616153596451405808351, 7.78435052129829890563566688100, 8.516935475212449492297141730528, 10.06641991648732613249148363644, 10.58979053702150961152078319873