L(s) = 1 | + (0.5 − 0.866i)2-s + (0.707 + 0.707i)3-s + (−0.499 − 0.866i)4-s + (0.965 − 0.258i)6-s − 0.999·8-s + 1.00i·9-s + 1.73·11-s + (0.258 − 0.965i)12-s + (−0.5 + 0.866i)16-s + (−0.965 + 1.67i)17-s + (0.866 + 0.500i)18-s + (0.258 + 0.448i)19-s + (0.866 − 1.49i)22-s + (−0.707 − 0.707i)24-s + 25-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (0.707 + 0.707i)3-s + (−0.499 − 0.866i)4-s + (0.965 − 0.258i)6-s − 0.999·8-s + 1.00i·9-s + 1.73·11-s + (0.258 − 0.965i)12-s + (−0.5 + 0.866i)16-s + (−0.965 + 1.67i)17-s + (0.866 + 0.500i)18-s + (0.258 + 0.448i)19-s + (0.866 − 1.49i)22-s + (−0.707 − 0.707i)24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.094018988\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.094018988\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
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
−8.762570796433351614779832619424, −8.537855052142512489919438914542, −7.15524333626202189405279640299, −6.31742942354063607469223819964, −5.53302050823761388101278900615, −4.50625185085525776241102852364, −3.93130531193562035192135506195, −3.45282127153699121431134588537, −2.25448164432190024823871227298, −1.48221542707436090134328196600,
1.11187407054174843055447514559, 2.59903637252036010536505890068, 3.26809601739081937249974249017, 4.28226555245638174831383771609, 4.88436114914815448523240177162, 6.13061208419944229374041641935, 6.70277454270339461556106839456, 7.08603293496797177355583903399, 7.919676125112395341459817480132, 8.709250308519477139633156070374