L(s) = 1 | + (−0.0560 − 0.209i)2-s + (−5.01 + 1.34i)3-s + (6.88 − 3.97i)4-s + (10.0 − 10.0i)5-s + (0.562 + 0.973i)6-s + (17.5 + 4.68i)7-s + (−2.44 − 2.44i)8-s + (23.3 − 13.5i)9-s + (−2.66 − 1.53i)10-s + (−39.2 + 10.5i)11-s + (−29.2 + 29.2i)12-s + (−30.4 − 35.6i)13-s − 3.92i·14-s + (−36.8 + 63.9i)15-s + (31.4 − 54.4i)16-s + (58.9 + 102. i)17-s + ⋯ |
L(s) = 1 | + (−0.0198 − 0.0739i)2-s + (−0.965 + 0.259i)3-s + (0.860 − 0.497i)4-s + (0.898 − 0.898i)5-s + (0.0382 + 0.0662i)6-s + (0.944 + 0.253i)7-s + (−0.107 − 0.107i)8-s + (0.865 − 0.500i)9-s + (−0.0841 − 0.0486i)10-s + (−1.07 + 0.288i)11-s + (−0.702 + 0.703i)12-s + (−0.649 − 0.760i)13-s − 0.0748i·14-s + (−0.634 + 1.10i)15-s + (0.491 − 0.850i)16-s + (0.840 + 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.25814 - 0.372859i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25814 - 0.372859i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (5.01 - 1.34i)T \) |
| 13 | \( 1 + (30.4 + 35.6i)T \) |
good | 2 | \( 1 + (0.0560 + 0.209i)T + (-6.92 + 4i)T^{2} \) |
| 5 | \( 1 + (-10.0 + 10.0i)T - 125iT^{2} \) |
| 7 | \( 1 + (-17.5 - 4.68i)T + (297. + 171.5i)T^{2} \) |
| 11 | \( 1 + (39.2 - 10.5i)T + (1.15e3 - 665.5i)T^{2} \) |
| 17 | \( 1 + (-58.9 - 102. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (8.33 - 31.0i)T + (-5.94e3 - 3.42e3i)T^{2} \) |
| 23 | \( 1 + (37.9 - 65.7i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (106. + 61.4i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-49.4 - 49.4i)T + 2.97e4iT^{2} \) |
| 37 | \( 1 + (-33.4 - 124. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (77.5 + 289. i)T + (-5.96e4 + 3.44e4i)T^{2} \) |
| 43 | \( 1 + (311. - 180. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-324. - 324. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 - 207. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-43.3 + 161. i)T + (-1.77e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-52.8 - 91.4i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-618. + 165. i)T + (2.60e5 - 1.50e5i)T^{2} \) |
| 71 | \( 1 + (234. + 62.9i)T + (3.09e5 + 1.78e5i)T^{2} \) |
| 73 | \( 1 + (305. - 305. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 452.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-646. + 646. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + (-986. + 264. i)T + (6.10e5 - 3.52e5i)T^{2} \) |
| 97 | \( 1 + (-269. + 1.00e3i)T + (-7.90e5 - 4.56e5i)T^{2} \) |
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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
−15.68139246315839793132110980687, −14.77414838049855334865047640692, −12.93131708495094895990059898120, −12.00466268218247923485393711063, −10.62639313803619017304384924928, −9.883408365358220245450714258524, −7.83145299332087038763180505605, −5.83043477303978955067906718605, −5.16637670856344330536243102689, −1.62450492066975803535189858368,
2.31922111904246585657170402189, 5.25614053265689892993759536993, 6.73012688382061765124456607135, 7.67201640370903601321465996121, 10.12452176833515934088169823336, 11.08913505629494214671305657825, 11.96012474330312910311089248850, 13.47562405688260104103881105839, 14.67744048085385483153828911654, 16.13970255442845028640727985502