L(s) = 1 | + (5.65 − 5.65i)3-s + (−54.5 + 12.2i)5-s + (−23.8 − 23.8i)7-s + 178. i·9-s − 218. i·11-s + (152. + 152. i)13-s + (−239. + 377. i)15-s + (318. − 318. i)17-s + 2.45e3·19-s − 270.·21-s + (512. − 512. i)23-s + (2.82e3 − 1.33e3i)25-s + (2.38e3 + 2.38e3i)27-s − 5.94e3i·29-s + 3.06e3i·31-s + ⋯ |
L(s) = 1 | + (0.362 − 0.362i)3-s + (−0.975 + 0.218i)5-s + (−0.184 − 0.184i)7-s + 0.736i·9-s − 0.543i·11-s + (0.250 + 0.250i)13-s + (−0.274 + 0.433i)15-s + (0.267 − 0.267i)17-s + 1.56·19-s − 0.133·21-s + (0.201 − 0.201i)23-s + (0.904 − 0.426i)25-s + (0.630 + 0.630i)27-s − 1.31i·29-s + 0.572i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.806080476\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.806080476\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (54.5 - 12.2i)T \) |
good | 3 | \( 1 + (-5.65 + 5.65i)T - 243iT^{2} \) |
| 7 | \( 1 + (23.8 + 23.8i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 218. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-152. - 152. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-318. + 318. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 2.45e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-512. + 512. i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 5.94e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 3.06e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.31e3 + 1.31e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 6.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.90e3 + 1.90e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-8.04e3 - 8.04e3i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (8.09e3 + 8.09e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 4.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-4.16e4 - 4.16e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.37e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-9.33e3 - 9.33e3i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 8.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-7.58e4 + 7.58e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.86e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (9.77e4 - 9.77e4i)T - 8.58e9iT^{2} \) |
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
−11.80350794735021982765881867929, −11.12718439927436337684259029402, −9.935462037582768884575881463466, −8.586734941087769021699272185783, −7.74936700175342107194794822852, −6.91353568035158847200317822707, −5.33366951922258031015308522036, −3.88270028593895375785471971191, −2.69659919423042474862827350200, −0.834733175852649469761344399315,
0.926040343453122724688755757587, 3.08074630156448820866213041123, 4.00631227380335724057812595369, 5.35728210953460660045263545914, 6.90298699449715797560541798358, 7.931561991144017590773973787556, 9.017904158329035108487944529048, 9.832268886634227838958716070348, 11.13643463647634797813424421830, 12.09589893594095713008423165783