| L(s) = 1 | − 1.73i·3-s − 4i·7-s − 2.99·9-s − 6.92·13-s + 3.46i·19-s − 6.92·21-s + 5·25-s + 5.19i·27-s − 4i·31-s − 6.92·37-s + 11.9i·39-s − 10.3i·43-s − 9·49-s + 5.99·57-s − 6.92·61-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s − 1.51i·7-s − 0.999·9-s − 1.92·13-s + 0.794i·19-s − 1.51·21-s + 25-s + 0.999i·27-s − 0.718i·31-s − 1.13·37-s + 1.92i·39-s − 1.58i·43-s − 1.28·49-s + 0.794·57-s − 0.887·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.813042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.813042i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.92T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
| 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
−10.08304771043510332464102829024, −8.969556862088549867935975870004, −7.83900220098544426041903743922, −7.30301023703556145048577597769, −6.72099303086368314441288469245, −5.46219528296260894330655481783, −4.42640012051705735366309822773, −3.16099152566024428628998385761, −1.85713670464941388887287877980, −0.38656070946212232415427935165,
2.39696351525790347273086422176, 3.10799123540436138661274202638, 4.76434014709047746669157101541, 5.09604929990128445135117146477, 6.17712434416083069124374520863, 7.32369206557231966084609707164, 8.524454835194457889063570509421, 9.114967995855579530715927878871, 9.785713111280044907441845975572, 10.62774442227213058915915793898
