| L(s) = 1 | + (−4.97 + 1.81i)2-s + (−3.67 + 3.67i)3-s + (15.3 − 12.9i)4-s + (−1.82 − 10.3i)5-s + (11.6 − 24.9i)6-s + (−5.92 − 4.96i)7-s + (−31.9 + 55.3i)8-s + (0.0364 − 26.9i)9-s + (27.8 + 48.2i)10-s + (2.99 − 16.9i)11-s + (−9.15 + 103. i)12-s + (−67.4 − 24.5i)13-s + (38.4 + 14.0i)14-s + (44.7 + 31.3i)15-s + (30.9 − 175. i)16-s + (−23.3 − 40.4i)17-s + ⋯ |
| L(s) = 1 | + (−1.76 + 0.640i)2-s + (−0.707 + 0.706i)3-s + (1.92 − 1.61i)4-s + (−0.163 − 0.925i)5-s + (0.792 − 1.69i)6-s + (−0.319 − 0.268i)7-s + (−1.41 + 2.44i)8-s + (0.00134 − 0.999i)9-s + (0.880 + 1.52i)10-s + (0.0820 − 0.465i)11-s + (−0.220 + 2.49i)12-s + (−1.43 − 0.523i)13-s + (0.734 + 0.267i)14-s + (0.769 + 0.539i)15-s + (0.483 − 2.74i)16-s + (−0.333 − 0.577i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0270 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0270 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.142034 - 0.138241i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.142034 - 0.138241i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.67 - 3.67i)T \) |
| good | 2 | \( 1 + (4.97 - 1.81i)T + (6.12 - 5.14i)T^{2} \) |
| 5 | \( 1 + (1.82 + 10.3i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (5.92 + 4.96i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (-2.99 + 16.9i)T + (-1.25e3 - 455. i)T^{2} \) |
| 13 | \( 1 + (67.4 + 24.5i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (23.3 + 40.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (21.3 - 36.9i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-36.3 + 30.5i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (139. - 50.8i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (120. - 101. i)T + (5.17e3 - 2.93e4i)T^{2} \) |
| 37 | \( 1 + (-35.3 - 61.2i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (219. + 79.8i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-71.5 + 405. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-327. - 275. i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 - 43.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + (57.9 + 328. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-292. - 245. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (230. + 83.9i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-403. - 698. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-175. + 303. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-9.99e2 + 363. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (959. - 349. i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-320. + 555. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-60.5 + 343. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.75643505204580486783296742661, −15.98113181031536309994430315961, −14.85529326700521138282685391270, −12.26822145856954319916109403850, −10.82066414227898608831399278589, −9.736481879280681309870817747118, −8.720406020343865834160935969782, −7.06965634178584925744342632286, −5.34841752229001997160381366933, −0.34709593077116122194215721108,
2.30508093884605609361262026822, 6.73368558866435084205526898646, 7.60684881628411948000616370884, 9.434471499082734571318383716230, 10.69664984093828308744753164082, 11.62975354132174677959807218031, 12.71722549467447083248042933156, 15.08042784199882822683271098189, 16.67368131288198248801047995580, 17.42554496482041569484900563876
