L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − i·7-s + 0.999i·8-s − i·9-s + (1.36 + 1.36i)11-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)18-s + (0.499 + 1.86i)22-s − 1.73i·23-s + (0.866 − 0.499i)28-s + (−0.366 + 0.366i)29-s + (−0.866 + 0.499i)32-s + (0.866 − 0.499i)36-s + (0.366 + 0.366i)37-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − i·7-s + 0.999i·8-s − i·9-s + (1.36 + 1.36i)11-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)18-s + (0.499 + 1.86i)22-s − 1.73i·23-s + (0.866 − 0.499i)28-s + (−0.366 + 0.366i)29-s + (−0.866 + 0.499i)32-s + (0.866 − 0.499i)36-s + (0.366 + 0.366i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.197944649\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.197944649\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + 1.73iT - T^{2} \) |
| 29 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 71 | \( 1 - iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−9.076397064212337810464782758725, −8.074473144195480546451254048274, −7.30043967472874582135187968210, −6.48906245454637751763365129263, −6.45674167532935703206983975922, −5.00165299974821641126404126793, −4.20735199902662355723184962481, −3.88005649991969409995182781396, −2.71004469674087311632510074625, −1.37982034474242085674155508620,
1.38738079779154331707362038062, 2.34159704986618946046183999733, 3.31411428146883632235041803587, 4.02493009463287783711916227672, 5.10844725217543607380564597809, 5.76541506615932178819909689287, 6.22721697444205374445037245935, 7.30570044293596974159547207361, 8.162770092536624388773037166445, 9.226407627569600003999880979658