L(s) = 1 | + (1.41 − 1.73i)5-s − 2.82i·7-s + 4.89·11-s − 4.89i·13-s − 3.46i·17-s − 6.92·19-s − 4i·23-s + (−0.999 − 4.89i)25-s − 8.48·29-s + 6.92·31-s + (−4.89 − 4.00i)35-s + 4.89i·37-s − 5.65·41-s + 11.3i·43-s + 4i·47-s + ⋯ |
L(s) = 1 | + (0.632 − 0.774i)5-s − 1.06i·7-s + 1.47·11-s − 1.35i·13-s − 0.840i·17-s − 1.58·19-s − 0.834i·23-s + (−0.199 − 0.979i)25-s − 1.57·29-s + 1.24·31-s + (−0.828 − 0.676i)35-s + 0.805i·37-s − 0.883·41-s + 1.72i·43-s + 0.583i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.867915402\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.867915402\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 11.3iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 9.79iT - 97T^{2} \) |
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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
−8.416315870869815741674764409848, −7.954022257748036167793114243582, −6.75611741061739692002832053871, −6.39117179450170122509694699726, −5.36367624693648025410302448380, −4.50441236453546550459133609825, −3.91214362589889189425339840073, −2.72184571604097308270476047379, −1.46454691172009501155954232993, −0.58507301106975968808572397631,
1.84655694573903564217453523229, 2.09432261160031714719642572707, 3.55132541829580070319121109861, 4.14761073633231854018547451272, 5.37868444365466965113824141088, 6.22630627464340667194178303456, 6.52417569809397941679012322576, 7.35692650658368800790456308079, 8.610558210517690250681359562841, 9.014525880521578001873898347251