| L(s) = 1 | + (3.28 − 9.03i)3-s + (7.51 + 42.6i)5-s + (−18.8 + 32.6i)7-s + (−8.81 − 7.39i)9-s + (103. + 180. i)11-s + (−76.3 − 209. i)13-s + (409. + 72.2i)15-s + (361. − 303. i)17-s + (−54.7 + 356. i)19-s + (233. + 278. i)21-s + (−61.0 + 346. i)23-s + (−1.17e3 + 426. i)25-s + (578. − 334. i)27-s + (−524. + 625. i)29-s + (162. + 93.8i)31-s + ⋯ |
| L(s) = 1 | + (0.365 − 1.00i)3-s + (0.300 + 1.70i)5-s + (−0.385 + 0.667i)7-s + (−0.108 − 0.0912i)9-s + (0.859 + 1.48i)11-s + (−0.452 − 1.24i)13-s + (1.82 + 0.321i)15-s + (1.25 − 1.05i)17-s + (−0.151 + 0.988i)19-s + (0.529 + 0.630i)21-s + (−0.115 + 0.654i)23-s + (−1.87 + 0.682i)25-s + (0.794 − 0.458i)27-s + (−0.623 + 0.743i)29-s + (0.169 + 0.0976i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.75352 + 0.659757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75352 + 0.659757i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (54.7 - 356. i)T \) |
| good | 3 | \( 1 + (-3.28 + 9.03i)T + (-62.0 - 52.0i)T^{2} \) |
| 5 | \( 1 + (-7.51 - 42.6i)T + (-587. + 213. i)T^{2} \) |
| 7 | \( 1 + (18.8 - 32.6i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-103. - 180. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (76.3 + 209. i)T + (-2.18e4 + 1.83e4i)T^{2} \) |
| 17 | \( 1 + (-361. + 303. i)T + (1.45e4 - 8.22e4i)T^{2} \) |
| 23 | \( 1 + (61.0 - 346. i)T + (-2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (524. - 625. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-162. - 93.8i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 914. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-819. + 2.25e3i)T + (-2.16e6 - 1.81e6i)T^{2} \) |
| 43 | \( 1 + (232. + 1.31e3i)T + (-3.21e6 + 1.16e6i)T^{2} \) |
| 47 | \( 1 + (-670. - 563. i)T + (8.47e5 + 4.80e6i)T^{2} \) |
| 53 | \( 1 + (4.53e3 + 798. i)T + (7.41e6 + 2.69e6i)T^{2} \) |
| 59 | \( 1 + (-2.50e3 - 2.98e3i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-173. + 984. i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (-1.69e3 + 2.01e3i)T + (-3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (2.94e3 - 519. i)T + (2.38e7 - 8.69e6i)T^{2} \) |
| 73 | \( 1 + (-455. - 165. i)T + (2.17e7 + 1.82e7i)T^{2} \) |
| 79 | \( 1 + (-847. + 2.32e3i)T + (-2.98e7 - 2.50e7i)T^{2} \) |
| 83 | \( 1 + (-1.81e3 + 3.14e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (2.61e3 + 7.18e3i)T + (-4.80e7 + 4.03e7i)T^{2} \) |
| 97 | \( 1 + (-125. - 149. i)T + (-1.53e7 + 8.71e7i)T^{2} \) |
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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
−14.13627941925447124312901008549, −12.67907674562236177951862469348, −12.01790917079753619654275936911, −10.40291042517504125577706679517, −9.571725053630966218058200503472, −7.56690472347207786595081608645, −7.11686953461978870984705643661, −5.77104178413739412406485978912, −3.19370113143134254395353938035, −2.00039940038513500409733642354,
1.01990288568715371798999246517, 3.75883609399962732992675493452, 4.68424732197032783733421032036, 6.25421683220317309502014911066, 8.322943743129004904942285596394, 9.203674152723664632207437905659, 9.903233613890810505059011953822, 11.42961427788182976811805130321, 12.66899563390577202094927605225, 13.68493215364886286928807668376
