L(s) = 1 | − 4.27i·3-s + 7.92·5-s + 5.75·7-s − 9.24·9-s + 16.5·11-s − 7.95i·13-s − 33.8i·15-s − 18.1·17-s + (0.595 + 18.9i)19-s − 24.5i·21-s + 7.02·23-s + 37.7·25-s + 1.04i·27-s − 30.4i·29-s + 34.9i·31-s + ⋯ |
L(s) = 1 | − 1.42i·3-s + 1.58·5-s + 0.822·7-s − 1.02·9-s + 1.50·11-s − 0.612i·13-s − 2.25i·15-s − 1.06·17-s + (0.0313 + 0.999i)19-s − 1.17i·21-s + 0.305·23-s + 1.50·25-s + 0.0387i·27-s − 1.05i·29-s + 1.12i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0313 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.233562429\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.233562429\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-0.595 - 18.9i)T \) |
good | 3 | \( 1 + 4.27iT - 9T^{2} \) |
| 5 | \( 1 - 7.92T + 25T^{2} \) |
| 7 | \( 1 - 5.75T + 49T^{2} \) |
| 11 | \( 1 - 16.5T + 121T^{2} \) |
| 13 | \( 1 + 7.95iT - 169T^{2} \) |
| 17 | \( 1 + 18.1T + 289T^{2} \) |
| 23 | \( 1 - 7.02T + 529T^{2} \) |
| 29 | \( 1 + 30.4iT - 841T^{2} \) |
| 31 | \( 1 - 34.9iT - 961T^{2} \) |
| 37 | \( 1 + 58.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 40.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 12.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 88.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 62.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 51.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 68.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 82.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 21.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 57.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 88.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 0.720T + 6.88e3T^{2} \) |
| 89 | \( 1 + 30.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 169. iT - 9.40e3T^{2} \) |
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
−9.182244716202878699152621782125, −8.549091598975789419842991984704, −7.58780526743342855548431647701, −6.72918019345242431524749333887, −6.14081120378019798075659942890, −5.45038367939128542496712639632, −4.16344113634485882388006499596, −2.53368760194414098337303679142, −1.73001906677043213735925019252, −1.09088548609037308245604612564,
1.42327820629775891484352913849, 2.45127091884934280105056731604, 3.85082099007628164929697961481, 4.65914000604621253765255495382, 5.27917673533278026685722418522, 6.33013623920618025346495397252, 6.99728056188029053423974945213, 8.743972328461705523992824062425, 9.029491474359277581206378311414, 9.650207470967527059191006791631