L(s) = 1 | + (−1.88 + 4.84i)3-s + (−3.20 − 5.55i)5-s + (14.8 − 11.1i)7-s + (−19.8 − 18.2i)9-s + (−16.2 − 9.39i)11-s − 3.82i·13-s + (32.9 − 5.03i)15-s + (8.12 − 14.0i)17-s + (129. − 74.9i)19-s + (25.8 + 92.6i)21-s + (109. − 63.3i)23-s + (41.9 − 72.6i)25-s + (126. − 61.6i)27-s + 168. i·29-s + (−43.4 − 25.0i)31-s + ⋯ |
L(s) = 1 | + (−0.363 + 0.931i)3-s + (−0.286 − 0.496i)5-s + (0.799 − 0.600i)7-s + (−0.735 − 0.676i)9-s + (−0.446 − 0.257i)11-s − 0.0816i·13-s + (0.567 − 0.0867i)15-s + (0.115 − 0.200i)17-s + (1.56 − 0.904i)19-s + (0.268 + 0.963i)21-s + (0.994 − 0.573i)23-s + (0.335 − 0.581i)25-s + (0.898 − 0.439i)27-s + 1.07i·29-s + (−0.251 − 0.145i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.28824 - 0.432810i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28824 - 0.432810i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.88 - 4.84i)T \) |
| 7 | \( 1 + (-14.8 + 11.1i)T \) |
good | 5 | \( 1 + (3.20 + 5.55i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (16.2 + 9.39i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 3.82iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-8.12 + 14.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-129. + 74.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-109. + 63.3i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 168. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (43.4 + 25.0i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-24.7 - 42.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 19.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 5.14T + 7.95e4T^{2} \) |
| 47 | \( 1 + (308. + 534. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (242. + 140. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-274. + 475. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (507. - 293. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (378. - 655. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 351. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (530. + 306. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-505. - 876. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (178. + 309. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 228. iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95858799949327933602199947842, −11.16813843239828940231662774163, −10.37047117373544986551282845634, −9.207251436297556968754451465982, −8.261173531743422514430369341749, −6.99718180335774228602247432723, −5.27336270193517097235800867622, −4.68797959544357091324562630438, −3.23106534313348678251425462462, −0.73295992446445225408632314504,
1.46899550709270895176496163802, 2.99357642135131261903377520463, 5.02383693531721003902639074034, 6.00660860291456229200975039009, 7.42010334069280344457492617509, 7.919095733723287740287210513184, 9.316138004374704333676440552976, 10.78286146292860318158031609622, 11.54662313916055645197597465781, 12.28134080681196030368465097298