L(s) = 1 | + (−2.37 + 0.636i)2-s + (3.50 − 2.02i)4-s + (0.498 − 0.498i)5-s + (−2.62 + 0.304i)7-s + (−3.56 + 3.56i)8-s + (−0.866 + 1.50i)10-s + (−0.184 − 0.688i)11-s + (−3.17 − 1.70i)13-s + (6.05 − 2.39i)14-s + (2.15 − 3.73i)16-s + (2.27 + 3.93i)17-s + (3.24 + 0.870i)19-s + (0.739 − 2.75i)20-s + (0.876 + 1.51i)22-s + (1.67 + 0.964i)23-s + ⋯ |
L(s) = 1 | + (−1.68 + 0.450i)2-s + (1.75 − 1.01i)4-s + (0.222 − 0.222i)5-s + (−0.993 + 0.114i)7-s + (−1.26 + 1.26i)8-s + (−0.274 + 0.474i)10-s + (−0.0556 − 0.207i)11-s + (−0.881 − 0.471i)13-s + (1.61 − 0.640i)14-s + (0.538 − 0.932i)16-s + (0.551 + 0.954i)17-s + (0.745 + 0.199i)19-s + (0.165 − 0.616i)20-s + (0.186 + 0.323i)22-s + (0.348 + 0.201i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.569643 + 0.0494688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.569643 + 0.0494688i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.62 - 0.304i)T \) |
| 13 | \( 1 + (3.17 + 1.70i)T \) |
good | 2 | \( 1 + (2.37 - 0.636i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-0.498 + 0.498i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.184 + 0.688i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.27 - 3.93i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.24 - 0.870i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.67 - 0.964i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.185 - 0.322i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.53 + 3.53i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.545 + 2.03i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (3.11 + 11.6i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.38 + 3.68i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.55 - 3.55i)T + 47iT^{2} \) |
| 53 | \( 1 - 4.97T + 53T^{2} \) |
| 59 | \( 1 + (-1.03 + 3.85i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-10.0 + 5.81i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12.5 - 3.37i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (2.10 - 7.86i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.608 - 0.608i)T + 73iT^{2} \) |
| 79 | \( 1 - 9.81T + 79T^{2} \) |
| 83 | \( 1 + (-2.25 + 2.25i)T - 83iT^{2} \) |
| 89 | \( 1 + (-17.5 + 4.70i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-8.73 - 2.34i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.10507959172893057923066899128, −9.317300256948974625811281146285, −8.762761108920426521036251848653, −7.68104055565532267952837955078, −7.18450859516869164787278194824, −6.09339700976921552796981602281, −5.41899922314916882538429418917, −3.55867648794578623353993014848, −2.23023207339111775809716692651, −0.73530953186242604821900311908,
0.819703057830718746825499853116, 2.42398303522582079708055719387, 3.13682397299642941694065045434, 4.81812013326498491671345688603, 6.30825312475011111312048586384, 7.08769454402851575895812501817, 7.69470290237946763436311483230, 8.816556495876871753196796007866, 9.541683301619321167581054688154, 9.990033590488191675836954298644