L(s) = 1 | + (0.764 + 1.84i)2-s + (−0.130 − 0.130i)3-s + (−2.83 + 2.82i)4-s + (4.38 + 2.39i)5-s + (0.141 − 0.341i)6-s + (−1.59 − 1.59i)7-s + (−7.38 − 3.07i)8-s − 8.96i·9-s + (−1.07 + 9.94i)10-s − 11.9i·11-s + (0.739 + 0.000715i)12-s + (9.59 + 9.59i)13-s + (1.73 − 4.17i)14-s + (−0.260 − 0.887i)15-s + (0.0309 − 15.9i)16-s + (0.857 + 0.857i)17-s + ⋯ |
L(s) = 1 | + (0.382 + 0.924i)2-s + (−0.0435 − 0.0435i)3-s + (−0.707 + 0.706i)4-s + (0.877 + 0.479i)5-s + (0.0236 − 0.0569i)6-s + (−0.228 − 0.228i)7-s + (−0.923 − 0.384i)8-s − 0.996i·9-s + (−0.107 + 0.994i)10-s − 1.08i·11-s + (0.0616 + 5.96e−5i)12-s + (0.738 + 0.738i)13-s + (0.123 − 0.298i)14-s + (−0.0173 − 0.0591i)15-s + (0.00193 − 0.999i)16-s + (0.0504 + 0.0504i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02481 + 0.723894i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02481 + 0.723894i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.764 - 1.84i)T \) |
| 5 | \( 1 + (-4.38 - 2.39i)T \) |
good | 3 | \( 1 + (0.130 + 0.130i)T + 9iT^{2} \) |
| 7 | \( 1 + (1.59 + 1.59i)T + 49iT^{2} \) |
| 11 | \( 1 + 11.9iT - 121T^{2} \) |
| 13 | \( 1 + (-9.59 - 9.59i)T + 169iT^{2} \) |
| 17 | \( 1 + (-0.857 - 0.857i)T + 289iT^{2} \) |
| 19 | \( 1 + 20.5T + 361T^{2} \) |
| 23 | \( 1 + (22.1 - 22.1i)T - 529iT^{2} \) |
| 29 | \( 1 + 27.3T + 841T^{2} \) |
| 31 | \( 1 - 40.0T + 961T^{2} \) |
| 37 | \( 1 + (1.57 - 1.57i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 37.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-49.2 - 49.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (34.0 + 34.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (28.8 + 28.8i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 92.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 4.82iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (54.6 - 54.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 59.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-34.1 + 34.1i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 96.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (63.6 + 63.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 3.68iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-46.0 - 46.0i)T + 9.40e3iT^{2} \) |
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
−16.15782466273341269618485461370, −14.91717294191074201435213068076, −13.91054606355536525518339414497, −13.11045069474501504299133055615, −11.50938703097887808951199414735, −9.752797556818390506871719664388, −8.542420080536914006499518088814, −6.67219267800265732205754429287, −5.91209662360198874010712963564, −3.66439460205249472334318488960,
2.17195988595542199136343272521, 4.60535356353336603120044404360, 5.98032687136995174376314663302, 8.467971962474587957291425221452, 9.880468774796374623934585675903, 10.71783751886155069485603915039, 12.39992109809274126303292069302, 13.14363567830613533989710181117, 14.18988066028764433539213372045, 15.56023172695676305770310837546