L(s) = 1 | + 2.82·5-s − i·7-s − 1.41i·11-s − 2i·13-s − 2.82i·17-s − 4·19-s − 1.41·23-s + 3.00·25-s + 1.41·29-s − 2.82i·35-s − 10i·37-s − 5.65i·41-s − 2·43-s − 2.82·47-s − 49-s + ⋯ |
L(s) = 1 | + 1.26·5-s − 0.377i·7-s − 0.426i·11-s − 0.554i·13-s − 0.685i·17-s − 0.917·19-s − 0.294·23-s + 0.600·25-s + 0.262·29-s − 0.478i·35-s − 1.64i·37-s − 0.883i·41-s − 0.304·43-s − 0.412·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.884484792\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.884484792\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 + 8.48iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 + 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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
−8.309064096979792902471737894869, −7.46076956882356859666521733744, −6.69427386633890933464922714377, −5.94220038570519448194916089165, −5.43257936317300005251271268971, −4.52248975144401655074347346632, −3.54786167550386909182670310327, −2.56013378497228691770130318864, −1.77698228163816288316211788801, −0.49595681400993365209837501461,
1.47887309262312143815719468700, 2.11047854104509655797735025317, 3.02552300158137984709908375922, 4.22706873670196751718779378085, 4.91713313088572057797105714482, 5.84872182338975117187854715848, 6.33841804191066238365394184724, 6.96748049986969442083375271466, 8.107402387682625284817163465704, 8.635183100566230240293227349718