L(s) = 1 | − 5.19i·7-s + (−1.5 − 0.866i)13-s + (−4 − 1.73i)19-s + (2.5 − 4.33i)25-s − 11·31-s + 12.1i·37-s + (10.5 − 6.06i)43-s − 20·49-s + (−6.5 + 11.2i)61-s + (2.5 − 4.33i)67-s + (−3.5 − 6.06i)73-s + (6.5 + 11.2i)79-s + (−4.5 + 7.79i)91-s + (−12 + 6.92i)97-s − 13·103-s + ⋯ |
L(s) = 1 | − 1.96i·7-s + (−0.416 − 0.240i)13-s + (−0.917 − 0.397i)19-s + (0.5 − 0.866i)25-s − 1.97·31-s + 1.99i·37-s + (1.60 − 0.924i)43-s − 2.85·49-s + (−0.832 + 1.44i)61-s + (0.305 − 0.529i)67-s + (−0.409 − 0.709i)73-s + (0.731 + 1.26i)79-s + (−0.471 + 0.817i)91-s + (−1.21 + 0.703i)97-s − 1.28·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7121164977\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7121164977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 5.19iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 11T + 31T^{2} \) |
| 37 | \( 1 - 12.1iT - 37T^{2} \) |
| 41 | \( 1 + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-10.5 + 6.06i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (12 - 6.92i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.333455060210173564403718504570, −7.58818215180578250457858881005, −7.00191865631041658241799674209, −6.36747841830732286112121533392, −5.19376121310791619633665344114, −4.35435377704753008912613443546, −3.80086640690622082520208224821, −2.69222629766550002500080713043, −1.36326145107949271972607416332, −0.22428242819769975415970190403,
1.82453035333257578499902791404, 2.47416338839528205052242930634, 3.50233521887085029830290537441, 4.59671503821827805616811907869, 5.57963501553589169832975105342, 5.86049083056930808109416911038, 6.91612203668815252784796650502, 7.75217281028595077847371960168, 8.608041226420124875172981369640, 9.217018669627786751824778410137