| L(s) = 1 | + (−0.772 − 0.561i)2-s + (−2.19 − 6.74i)4-s + (−11.0 + 1.36i)5-s + 12.4·7-s + (−4.45 + 13.7i)8-s + (9.34 + 5.17i)10-s + (−36.1 − 26.2i)11-s + (5.77 − 4.19i)13-s + (−9.64 − 7.00i)14-s + (−34.7 + 25.2i)16-s + (−8.19 + 25.2i)17-s + (−14.3 + 44.1i)19-s + (33.4 + 71.8i)20-s + (13.2 + 40.6i)22-s + (117. + 85.4i)23-s + ⋯ |
| L(s) = 1 | + (−0.273 − 0.198i)2-s + (−0.273 − 0.842i)4-s + (−0.992 + 0.121i)5-s + 0.674·7-s + (−0.196 + 0.605i)8-s + (0.295 + 0.163i)10-s + (−0.991 − 0.720i)11-s + (0.123 − 0.0894i)13-s + (−0.184 − 0.133i)14-s + (−0.542 + 0.394i)16-s + (−0.116 + 0.359i)17-s + (−0.173 + 0.532i)19-s + (0.374 + 0.802i)20-s + (0.127 + 0.393i)22-s + (1.06 + 0.775i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.414811 + 0.340707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.414811 + 0.340707i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (11.0 - 1.36i)T \) |
| good | 2 | \( 1 + (0.772 + 0.561i)T + (2.47 + 7.60i)T^{2} \) |
| 7 | \( 1 - 12.4T + 343T^{2} \) |
| 11 | \( 1 + (36.1 + 26.2i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-5.77 + 4.19i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (8.19 - 25.2i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (14.3 - 44.1i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-117. - 85.4i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-65.9 - 202. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (45.7 - 140. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (325. - 236. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-189. + 137. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 87.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-23.4 - 72.2i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (51.6 + 158. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (481. - 349. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-700. - 508. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (125. - 386. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (162. + 500. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (810. + 588. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (41.3 + 127. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (159. - 491. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (494. + 358. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (506. + 1.55e3i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
| 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
−11.74454536468285580008133400402, −10.84055421250375332403081259659, −10.44264979808940050668313068320, −8.873373814462581972049875274255, −8.273909475534934751851181265726, −7.10531379846958693779282382045, −5.60892689883364874971045937140, −4.72557184708144073732318697963, −3.17325497448269212319648500589, −1.30706147925499375927632352840,
0.26791698306881162624111726396, 2.66764836370677833630888216609, 4.14700704904629098165243374278, 4.96991353304327481228399499260, 6.91789173798483666514846255121, 7.71981188147373931640208224520, 8.389577433069772679439711267043, 9.389988761721365028674637134517, 10.78471277368549737240487768530, 11.61572572746988035481281232028
