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 + (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 + (−18.1 − 13.7i)18-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 + (0.531 − 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 + (−1.00 − 0.761i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.698573 + 0.164123i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.698573 + 0.164123i\) |
\(L(2)\) |
|
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 | \( 1 \) |
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 | \( 1 - 149.T + 9.40e3T^{2} \) |
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
−11.22809234999847040762157116965, −10.42219297233024035897860205433, −9.868701546121122765571373541939, −9.089064479467122617320847117465, −7.28742796022470551230842081172, −6.38752676768062143798355637348, −5.24573634063919200640776344882, −3.88620190720541997643853030273, −3.08011797297600168016539549896, −1.30165593428722335016829528314,
0.39065716315635361243385313878, 2.68014588804427720236458805973, 4.77089283749589369907728052037, 5.43876514375927257514435125039, 6.55866897057035480557951979891, 7.05943048157780258827524825755, 7.898725675774542855132189876123, 9.224306440404642023406822311245, 10.07265558416030311213574805228, 11.35447600092993361382210007156