| L(s) = 1 | + 5.23i·3-s + 2.23·5-s + 10.1i·7-s − 18.4·9-s − 14.4i·11-s + 11.5·13-s + 11.7i·15-s − 18.9·17-s + 12i·19-s − 53.3·21-s + 17.5i·23-s + 5.00·25-s − 49.3i·27-s + 8.83·29-s − 0.583i·31-s + ⋯ |
| L(s) = 1 | + 1.74i·3-s + 0.447·5-s + 1.45i·7-s − 2.04·9-s − 1.31i·11-s + 0.886·13-s + 0.780i·15-s − 1.11·17-s + 0.631i·19-s − 2.53·21-s + 0.765i·23-s + 0.200·25-s − 1.82i·27-s + 0.304·29-s − 0.0188i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.546243 + 1.31874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.546243 + 1.31874i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 3 | \( 1 - 5.23iT - 9T^{2} \) |
| 7 | \( 1 - 10.1iT - 49T^{2} \) |
| 11 | \( 1 + 14.4iT - 121T^{2} \) |
| 13 | \( 1 - 11.5T + 169T^{2} \) |
| 17 | \( 1 + 18.9T + 289T^{2} \) |
| 19 | \( 1 - 12iT - 361T^{2} \) |
| 23 | \( 1 - 17.5iT - 529T^{2} \) |
| 29 | \( 1 - 8.83T + 841T^{2} \) |
| 31 | \( 1 + 0.583iT - 961T^{2} \) |
| 37 | \( 1 - 32.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 71.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 4.65iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 22.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 63.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 30.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 65.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 92.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 41.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 136.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 81.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 86.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 30T + 7.92e3T^{2} \) |
| 97 | \( 1 - 119.T + 9.40e3T^{2} \) |
| 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
−13.13997061139935954710307478300, −11.58286469282065246030455602728, −11.04053943937606396353688547845, −9.926197003625063713020803276426, −8.925387414217908664443917543649, −8.543434312686635661227900751640, −6.00568516815998576092855254668, −5.51490483422965101716112464569, −4.03360635774704163372620882428, −2.76477361071610839094628931195,
0.945562377302948379937062281331, 2.32312926993369982151100376290, 4.41104086148112370037782446454, 6.26495504260815008770736639646, 6.98387411807777554495763009585, 7.74455711787102152472157550818, 9.015641402868030628603471397464, 10.45307229827901871090815641232, 11.38157973874728810887446039636, 12.63866889638314819798483076901
