| L(s) = 1 | + (11.6 + 2.04i)2-s + (14.4 + 22.8i)3-s + (70.6 + 25.7i)4-s + (2.84 + 3.38i)5-s + (121. + 294. i)6-s + (129. − 47.2i)7-s + (114. + 66.0i)8-s + (−311. + 659. i)9-s + (26.0 + 45.1i)10-s + (294. − 350. i)11-s + (434. + 1.98e3i)12-s + (−280. − 1.59e3i)13-s + (1.60e3 − 282. i)14-s + (−36.1 + 113. i)15-s + (−2.49e3 − 2.09e3i)16-s + (5.24e3 − 3.02e3i)17-s + ⋯ |
| L(s) = 1 | + (1.45 + 0.256i)2-s + (0.535 + 0.844i)3-s + (1.10 + 0.401i)4-s + (0.0227 + 0.0270i)5-s + (0.560 + 1.36i)6-s + (0.378 − 0.137i)7-s + (0.223 + 0.129i)8-s + (−0.427 + 0.904i)9-s + (0.0260 + 0.0451i)10-s + (0.220 − 0.263i)11-s + (0.251 + 1.14i)12-s + (−0.127 − 0.723i)13-s + (0.584 − 0.103i)14-s + (−0.0107 + 0.0336i)15-s + (−0.608 − 0.510i)16-s + (1.06 − 0.616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.31028 + 1.67712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.31028 + 1.67712i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-14.4 - 22.8i)T \) |
| good | 2 | \( 1 + (-11.6 - 2.04i)T + (60.1 + 21.8i)T^{2} \) |
| 5 | \( 1 + (-2.84 - 3.38i)T + (-2.71e3 + 1.53e4i)T^{2} \) |
| 7 | \( 1 + (-129. + 47.2i)T + (9.01e4 - 7.56e4i)T^{2} \) |
| 11 | \( 1 + (-294. + 350. i)T + (-3.07e5 - 1.74e6i)T^{2} \) |
| 13 | \( 1 + (280. + 1.59e3i)T + (-4.53e6 + 1.65e6i)T^{2} \) |
| 17 | \( 1 + (-5.24e3 + 3.02e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (2.35e3 - 4.08e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-340. + 935. i)T + (-1.13e8 - 9.51e7i)T^{2} \) |
| 29 | \( 1 + (4.57e4 + 8.07e3i)T + (5.58e8 + 2.03e8i)T^{2} \) |
| 31 | \( 1 + (-5.66e3 - 2.06e3i)T + (6.79e8 + 5.70e8i)T^{2} \) |
| 37 | \( 1 + (-4.38e4 - 7.58e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (3.66e4 - 6.46e3i)T + (4.46e9 - 1.62e9i)T^{2} \) |
| 43 | \( 1 + (-8.38e4 - 7.03e4i)T + (1.09e9 + 6.22e9i)T^{2} \) |
| 47 | \( 1 + (-5.51e4 - 1.51e5i)T + (-8.25e9 + 6.92e9i)T^{2} \) |
| 53 | \( 1 - 2.73e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (1.78e5 + 2.12e5i)T + (-7.32e9 + 4.15e10i)T^{2} \) |
| 61 | \( 1 + (-2.02e5 + 7.36e4i)T + (3.94e10 - 3.31e10i)T^{2} \) |
| 67 | \( 1 + (-2.85e4 - 1.61e5i)T + (-8.50e10 + 3.09e10i)T^{2} \) |
| 71 | \( 1 + (-1.02e5 + 5.90e4i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (7.24e4 - 1.25e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.41e4 - 1.93e5i)T + (-2.28e11 - 8.31e10i)T^{2} \) |
| 83 | \( 1 + (5.88e5 + 1.03e5i)T + (3.07e11 + 1.11e11i)T^{2} \) |
| 89 | \( 1 + (7.60e5 + 4.39e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-5.26e5 - 4.42e5i)T + (1.44e11 + 8.20e11i)T^{2} \) |
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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
−15.76887836572069937251174488945, −14.68920179771112881395634707477, −14.08090178713373444449844255322, −12.76907921015804222708294003180, −11.29698763121320544738311984112, −9.724448643370511685607731660306, −7.894959271517701860001073114441, −5.77688082749072298479025688425, −4.45170012689174182404932271455, −3.05222612250872817563810145065,
1.99637072181162429218637896278, 3.76004671722151066946541547288, 5.64204627999891233996424426402, 7.22516488450213490208971265422, 9.015597241984110175893309030931, 11.33714526789084140200619539076, 12.40052167581823575057507251902, 13.30508581415660928519561043320, 14.44038895645781815294503953417, 15.06641403655071195263781296299
