L(s) = 1 | − 1.73·3-s + 3.46i·5-s − 10i·7-s + 2.99·9-s + 13.8·11-s + 6.92i·13-s − 5.99i·15-s + 30·17-s + 6.92·19-s + 17.3i·21-s − 12i·23-s + 13.0·25-s − 5.19·27-s − 51.9i·29-s − 14i·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.692i·5-s − 1.42i·7-s + 0.333·9-s + 1.25·11-s + 0.532i·13-s − 0.399i·15-s + 1.76·17-s + 0.364·19-s + 0.824i·21-s − 0.521i·23-s + 0.520·25-s − 0.192·27-s − 1.79i·29-s − 0.451i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.35438 - 0.178308i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35438 - 0.178308i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
good | 5 | \( 1 - 3.46iT - 25T^{2} \) |
| 7 | \( 1 + 10iT - 49T^{2} \) |
| 11 | \( 1 - 13.8T + 121T^{2} \) |
| 13 | \( 1 - 6.92iT - 169T^{2} \) |
| 17 | \( 1 - 30T + 289T^{2} \) |
| 19 | \( 1 - 6.92T + 361T^{2} \) |
| 23 | \( 1 + 12iT - 529T^{2} \) |
| 29 | \( 1 + 51.9iT - 841T^{2} \) |
| 31 | \( 1 + 14iT - 961T^{2} \) |
| 37 | \( 1 - 55.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 62.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 84iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 17.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 62.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 96.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 48.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 60iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 86T + 5.32e3T^{2} \) |
| 79 | \( 1 + 38iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 13.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 78T + 7.92e3T^{2} \) |
| 97 | \( 1 - 62T + 9.40e3T^{2} \) |
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
−11.98239072444243991867961660574, −11.36546405680939044199359234602, −10.26309815864875178649925119467, −9.680174856624780606898863508562, −7.950296412175660009932137194185, −6.96092937799136587067319182356, −6.20734029505040151173074040493, −4.54009558063421533262989148563, −3.45944873318913548989537381029, −1.11549595265725183239867621065,
1.35159168805644506857560298816, 3.40541667637147456264740166295, 5.17300047385634553141767235245, 5.72103610479550161328848448541, 7.11116622380024298921202842592, 8.553196876841905510216378974810, 9.235212010804483720165140831616, 10.35687255083676153937471160253, 11.81458696434824255108966482269, 12.10742493481447135904757540719