L(s) = 1 | − 0.327·2-s + 1.73·3-s − 3.89·4-s − 3.00i·5-s − 0.568·6-s + 2.64i·7-s + 2.58·8-s + 2.99·9-s + 0.986i·10-s + 9.21i·11-s − 6.74·12-s + 0.0704·13-s − 0.867i·14-s − 5.20i·15-s + 14.7·16-s − 4.13i·17-s + ⋯ |
L(s) = 1 | − 0.163·2-s + 0.577·3-s − 0.973·4-s − 0.601i·5-s − 0.0946·6-s + 0.377i·7-s + 0.323·8-s + 0.333·9-s + 0.0986i·10-s + 0.838i·11-s − 0.561·12-s + 0.00542·13-s − 0.0619i·14-s − 0.347i·15-s + 0.920·16-s − 0.243i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.560376346\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.560376346\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73T \) |
| 7 | \( 1 - 2.64iT \) |
| 23 | \( 1 + (7.16 + 21.8i)T \) |
good | 2 | \( 1 + 0.327T + 4T^{2} \) |
| 5 | \( 1 + 3.00iT - 25T^{2} \) |
| 11 | \( 1 - 9.21iT - 121T^{2} \) |
| 13 | \( 1 - 0.0704T + 169T^{2} \) |
| 17 | \( 1 + 4.13iT - 289T^{2} \) |
| 19 | \( 1 - 18.2iT - 361T^{2} \) |
| 29 | \( 1 - 40.1T + 841T^{2} \) |
| 31 | \( 1 - 34.9T + 961T^{2} \) |
| 37 | \( 1 - 40.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 74.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 27.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 84.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 49.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 8.08iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 72.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 14.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 144. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 18.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 90.3iT - 9.40e3T^{2} \) |
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
−10.38920316445634968272111217784, −9.858010295621266482843551798891, −8.859930019122476288813726159821, −8.415067528646258206890668266982, −7.45510302994768051052256258655, −6.08751805964686856136375294734, −4.80790385138309970779637355429, −4.23878467782855150621143890626, −2.71058933715819173277580293551, −1.10089466840694210398888231799,
0.847636046673767857328155984068, 2.78014426908887406542524813283, 3.80111553911602560324948257697, 4.82727217293897515855699492682, 6.11788952473598747491057123376, 7.24931064811338889513189145534, 8.171036502059997110138397006821, 8.897264435748788072665557192179, 9.772497736511358586383052151927, 10.56648422281418981089449374009