L(s) = 1 | + (−2.95 − 0.5i)3-s + 8i·7-s + (8.5 + 2.95i)9-s + 17.7i·11-s + 2i·13-s + 17.7·17-s + 11·19-s + (4 − 23.6i)21-s − 35.4·23-s + (−23.6 − 13i)27-s + 35.4i·29-s + 46·31-s + (8.87 − 52.5i)33-s + 16i·37-s + (1 − 5.91i)39-s + ⋯ |
L(s) = 1 | + (−0.986 − 0.166i)3-s + 1.14i·7-s + (0.944 + 0.328i)9-s + 1.61i·11-s + 0.153i·13-s + 1.04·17-s + 0.578·19-s + (0.190 − 1.12i)21-s − 1.54·23-s + (−0.876 − 0.481i)27-s + 1.22i·29-s + 1.48·31-s + (0.268 − 1.59i)33-s + 0.432i·37-s + (0.0256 − 0.151i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.590i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9435559476\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9435559476\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.95 + 0.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 8iT - 49T^{2} \) |
| 11 | \( 1 - 17.7iT - 121T^{2} \) |
| 13 | \( 1 - 2iT - 169T^{2} \) |
| 17 | \( 1 - 17.7T + 289T^{2} \) |
| 19 | \( 1 - 11T + 361T^{2} \) |
| 23 | \( 1 + 35.4T + 529T^{2} \) |
| 29 | \( 1 - 35.4iT - 841T^{2} \) |
| 31 | \( 1 - 46T + 961T^{2} \) |
| 37 | \( 1 - 16iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 53.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 62iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 35.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 70.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16T + 3.72e3T^{2} \) |
| 67 | \( 1 - 113iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 106. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 101iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 68T + 6.24e3T^{2} \) |
| 83 | \( 1 - 17.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 53.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 22iT - 9.40e3T^{2} \) |
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
−10.00067276051882465218870789641, −9.251877593983138171155270187284, −8.124593743144683118818875373534, −7.30385181973812917947639822269, −6.51280482539555639546734414919, −5.56633841750845384883456934740, −5.02368519325453347194959849554, −3.96113236563399997059489270789, −2.43601108656783332642764654077, −1.43245983898348329395015517444,
0.36663172597538225547862270408, 1.22430071601473021767067985341, 3.17895047070393791056631547032, 4.02759186349882828533524722298, 4.96733628606506432308480739509, 6.06086552444017296486121887015, 6.38188353578158092494323675043, 7.77443217649436340053750940842, 8.056822830615782512057745565837, 9.637959768943008448873969058938