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 | \( 1 - 14T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.62774442227213058915915793898, −9.785713111280044907441845975572, −9.114967995855579530715927878871, −8.524454835194457889063570509421, −7.32369206557231966084609707164, −6.17712434416083069124374520863, −5.09604929990128445135117146477, −4.76434014709047746669157101541, −3.10799123540436138661274202638, −2.39696351525790347273086422176,
0.38656070946212232415427935165, 1.85713670464941388887287877980, 3.16099152566024428628998385761, 4.42640012051705735366309822773, 5.46219528296260894330655481783, 6.72099303086368314441288469245, 7.30301023703556145048577597769, 7.83900220098544426041903743922, 8.969556862088549867935975870004, 10.08304771043510332464102829024