L(s) = 1 | − 2.52i·2-s + (−2.68 − 1.33i)3-s − 2.37·4-s − 0.792i·5-s + (−3.37 + 6.78i)6-s + 6.74·7-s − 4.10i·8-s + (5.43 + 7.17i)9-s − 2·10-s + 3.31i·11-s + (6.37 + 3.16i)12-s + 9.48·13-s − 17.0i·14-s + (−1.05 + 2.12i)15-s − 19.8·16-s + 29.2i·17-s + ⋯ |
L(s) = 1 | − 1.26i·2-s + (−0.895 − 0.445i)3-s − 0.593·4-s − 0.158i·5-s + (−0.562 + 1.13i)6-s + 0.963·7-s − 0.513i·8-s + (0.603 + 0.797i)9-s − 0.200·10-s + 0.301i·11-s + (0.531 + 0.264i)12-s + 0.729·13-s − 1.21i·14-s + (−0.0705 + 0.141i)15-s − 1.24·16-s + 1.72i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.463658 - 0.748427i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.463658 - 0.748427i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.68 + 1.33i)T \) |
| 11 | \( 1 - 3.31iT \) |
good | 2 | \( 1 + 2.52iT - 4T^{2} \) |
| 5 | \( 1 + 0.792iT - 25T^{2} \) |
| 7 | \( 1 - 6.74T + 49T^{2} \) |
| 13 | \( 1 - 9.48T + 169T^{2} \) |
| 17 | \( 1 - 29.2iT - 289T^{2} \) |
| 19 | \( 1 + 26.2T + 361T^{2} \) |
| 23 | \( 1 + 26.9iT - 529T^{2} \) |
| 29 | \( 1 - 25.9iT - 841T^{2} \) |
| 31 | \( 1 + 2.86T + 961T^{2} \) |
| 37 | \( 1 + 2.39T + 1.36e3T^{2} \) |
| 41 | \( 1 - 17.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 12.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 41.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 89.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 14.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 63.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 63.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 4.55iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 53.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 55.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 65.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 14.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 149.T + 9.40e3T^{2} \) |
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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
−16.43020260714070259938855108752, −14.79763435069408702750368823623, −12.99422447532934789555777570233, −12.38141397789172462770821812042, −11.00486444852050747712896245855, −10.56179720247796994474597070107, −8.429983005306895403306275961389, −6.46829480048815188021083183211, −4.45329971210931960882440347768, −1.65654041489718236067155511163,
4.76262208178670543919733466368, 6.02035549250298108280615147416, 7.37441175036494008532991693648, 8.918944740889346157887122516605, 10.84036526998441555175128507846, 11.70968852981559669781787698033, 13.69327895828834475894436114817, 14.92712728876636484495397950383, 15.78590919128044371058837590570, 16.76417928530706413751207276223