L(s) = 1 | + (−5.67 − 3.27i)2-s + (−1.11 − 8.93i)3-s + (13.4 + 23.3i)4-s + (10.2 − 5.89i)5-s + (−22.9 + 54.3i)6-s + (26.6 − 46.1i)7-s − 71.6i·8-s + (−78.4 + 19.9i)9-s − 77.2·10-s + (108. + 62.4i)11-s + (193. − 146. i)12-s + (37.3 + 64.6i)13-s + (−302. + 174. i)14-s + (−64.0 − 84.5i)15-s + (−19.2 + 33.3i)16-s + 7.70i·17-s + ⋯ |
L(s) = 1 | + (−1.41 − 0.819i)2-s + (−0.124 − 0.992i)3-s + (0.841 + 1.45i)4-s + (0.408 − 0.235i)5-s + (−0.636 + 1.50i)6-s + (0.543 − 0.941i)7-s − 1.11i·8-s + (−0.969 + 0.246i)9-s − 0.772·10-s + (0.894 + 0.516i)11-s + (1.34 − 1.01i)12-s + (0.220 + 0.382i)13-s + (−1.54 + 0.890i)14-s + (−0.284 − 0.375i)15-s + (−0.0752 + 0.130i)16-s + 0.0266i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.300688 - 0.478629i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.300688 - 0.478629i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.11 + 8.93i)T \) |
good | 2 | \( 1 + (5.67 + 3.27i)T + (8 + 13.8i)T^{2} \) |
| 5 | \( 1 + (-10.2 + 5.89i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (-26.6 + 46.1i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-108. - 62.4i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-37.3 - 64.6i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 - 7.70iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 54.1T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-346. + 199. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (468. + 270. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-766. - 1.32e3i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.71e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.09e3 + 629. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.30e3 + 2.25e3i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (692. + 400. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 - 4.22e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (2.88e3 - 1.66e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-7.50 + 13.0i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (2.59e3 + 4.48e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 1.92e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 949.T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-118. + 206. i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (1.14e4 + 6.58e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 - 575. iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (7.56e3 - 1.30e4i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
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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
−19.93516897880718329856908314765, −18.89344862444938517608262025846, −17.48118799453427654505527479428, −17.04917694968387109330175563809, −13.96043669200756142979617557366, −12.15948300402956413529349975776, −10.77997602641107249984782359224, −8.935528831918611667198271517288, −7.25259808378858088326653754619, −1.43416954902843804324915323752,
5.97151479953988768215279134759, 8.497175320800506517831948013721, 9.657344127998523603231432929950, 11.24982781826406031449774784490, 14.61090413774006195410523688790, 15.68740555355982170701183770750, 16.98568305644517455714220455584, 17.98678855297121965151283662077, 19.39436813152561489970413526472, 21.12533244589575764919901964011