L(s) = 1 | + (−0.0580 − 0.0580i)2-s + (0.605 + 1.62i)3-s − 1.99i·4-s + (−1.89 + 1.18i)5-s + (0.0590 − 0.129i)6-s + (−2.14 + 2.14i)7-s + (−0.231 + 0.231i)8-s + (−2.26 + 1.96i)9-s + (0.178 + 0.0409i)10-s + 4.16i·11-s + (3.23 − 1.20i)12-s + (2.98 + 2.98i)13-s + 0.248·14-s + (−3.07 − 2.35i)15-s − 3.95·16-s + (1.86 + 1.86i)17-s + ⋯ |
L(s) = 1 | + (−0.0410 − 0.0410i)2-s + (0.349 + 0.936i)3-s − 0.996i·4-s + (−0.847 + 0.531i)5-s + (0.0241 − 0.0527i)6-s + (−0.810 + 0.810i)7-s + (−0.0819 + 0.0819i)8-s + (−0.755 + 0.654i)9-s + (0.0565 + 0.0129i)10-s + 1.25i·11-s + (0.933 − 0.348i)12-s + (0.828 + 0.828i)13-s + 0.0665·14-s + (−0.794 − 0.607i)15-s − 0.989·16-s + (0.451 + 0.451i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.537693 + 0.780131i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.537693 + 0.780131i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.605 - 1.62i)T \) |
| 5 | \( 1 + (1.89 - 1.18i)T \) |
| 19 | \( 1 + iT \) |
good | 2 | \( 1 + (0.0580 + 0.0580i)T + 2iT^{2} \) |
| 7 | \( 1 + (2.14 - 2.14i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.16iT - 11T^{2} \) |
| 13 | \( 1 + (-2.98 - 2.98i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.86 - 1.86i)T + 17iT^{2} \) |
| 23 | \( 1 + (-5.35 + 5.35i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 + 0.743T + 31T^{2} \) |
| 37 | \( 1 + (3.93 - 3.93i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.30iT - 41T^{2} \) |
| 43 | \( 1 + (-2.54 - 2.54i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.88 - 4.88i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.58 + 1.58i)T - 53iT^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 5.26T + 61T^{2} \) |
| 67 | \( 1 + (-1.58 + 1.58i)T - 67iT^{2} \) |
| 71 | \( 1 - 7.97iT - 71T^{2} \) |
| 73 | \( 1 + (-3.09 - 3.09i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.68iT - 79T^{2} \) |
| 83 | \( 1 + (0.803 - 0.803i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 + (-11.4 + 11.4i)T - 97iT^{2} \) |
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
−11.89114939269139608381880095672, −10.95520406711529417892187290090, −10.25900129916111001416731595835, −9.318406103941859922650504461143, −8.689455771483899011854940552410, −7.13199592192227650100442301703, −6.12396196333969127569506435254, −4.88285474657608464469674317590, −3.80234903631405778933432283792, −2.41116554642465742735739455322,
0.70338649911930054379001571758, 3.30326817771744340540805210614, 3.60630688011537614588106143878, 5.64892373577676048756711374744, 7.00857023993368491242035037982, 7.65926661644015912285488353023, 8.462866579425455725731100716852, 9.226626272362625703473685144394, 10.95528764298190895880353076022, 11.67432645701870383723775186676