L(s) = 1 | + (−1.22 + 0.707i)2-s + (−1.22 − 2.12i)5-s − 2.82i·8-s + (3 + 1.73i)10-s + (1.22 + 0.707i)11-s + 5.19i·13-s + (2.00 + 3.46i)16-s + (−2.44 + 4.24i)17-s + (−1.5 + 0.866i)19-s − 2·22-s + (−4.89 + 2.82i)23-s + (−0.499 + 0.866i)25-s + (−3.67 − 6.36i)26-s + 2.82i·29-s + (−1.5 − 0.866i)31-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)2-s + (−0.547 − 0.948i)5-s − 0.999i·8-s + (0.948 + 0.547i)10-s + (0.369 + 0.213i)11-s + 1.44i·13-s + (0.500 + 0.866i)16-s + (−0.594 + 1.02i)17-s + (−0.344 + 0.198i)19-s − 0.426·22-s + (−1.02 + 0.589i)23-s + (−0.0999 + 0.173i)25-s + (−0.720 − 1.24i)26-s + 0.525i·29-s + (−0.269 − 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.138123 + 0.391180i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.138123 + 0.391180i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.22 - 0.707i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.22 - 0.707i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.19iT - 13T^{2} \) |
| 17 | \( 1 + (2.44 - 4.24i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.89 - 2.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-6.12 - 10.6i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.44 + 1.41i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.44 - 4.24i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 1.73i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.07iT - 71T^{2} \) |
| 73 | \( 1 + (1.5 + 0.866i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + (-2.44 - 4.24i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10.3iT - 97T^{2} \) |
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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
−11.54928022173962771609589357002, −10.31108269757565157394605106320, −9.303543734152870678699257309795, −8.742070260118484208007730679059, −8.036525641795057330386969831790, −7.04536993746579495464528158901, −6.13431329155578752092901838130, −4.48891256186800905268486595711, −3.88273970107117388645674314634, −1.58126299528232681659487632190,
0.35547448536739301371692184644, 2.32465408528538339999183602548, 3.43788511376184162974423149266, 4.95541593508220290391223849567, 6.16455411294987780444178176634, 7.31126099192007951128571652518, 8.168184551169936499474265486714, 9.021385976860012840079671620768, 10.09680529417168074551189968713, 10.62914816981182780861302191962