L(s) = 1 | + (14.0 − 6.67i)3-s − 28.8·5-s + 44.6i·7-s + (153. − 188. i)9-s − 152. i·11-s + 578. i·13-s + (−407. + 192. i)15-s + 82.5i·17-s + 1.23e3·19-s + (297. + 628. i)21-s + 2.59e3·23-s − 2.28e3·25-s + (914. − 3.67e3i)27-s − 53.2·29-s + 4.03e3i·31-s + ⋯ |
L(s) = 1 | + (0.903 − 0.428i)3-s − 0.516·5-s + 0.344i·7-s + (0.633 − 0.773i)9-s − 0.379i·11-s + 0.948i·13-s + (−0.467 + 0.221i)15-s + 0.0692i·17-s + 0.783·19-s + (0.147 + 0.311i)21-s + 1.02·23-s − 0.732·25-s + (0.241 − 0.970i)27-s − 0.0117·29-s + 0.753i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.820624622\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.820624622\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-14.0 + 6.67i)T \) |
good | 5 | \( 1 + 28.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 44.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 152. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 578. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 82.5iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.23e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 53.2T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.03e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 3.92e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 4.85e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.45e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.01e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.82e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.40e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.34e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 3.70e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.06e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 3.28e4T + 8.58e9T^{2} \) |
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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
−10.41338627914239642880511879903, −9.176022465538597936674726155025, −8.778585210595626421414320910158, −7.61532150122911111194602755466, −6.98815002713722347594835798852, −5.73100723563212983230003151801, −4.28966884489925749741392911744, −3.31919037434093679064098450072, −2.18227632639131502351980367372, −0.866899051404468740509449749019,
0.892167344251910323733284869328, 2.49142352314212915642421790168, 3.52389741444785399193213622352, 4.44388260836839144133496387933, 5.57663180763959707263838847365, 7.22131444820114833697170562361, 7.73769465880066666722757216032, 8.722390007754111547581134343620, 9.660881018440933791216090341075, 10.41895264662640758817254725809