L(s) = 1 | − i·3-s + (−2 − i)5-s + 2i·7-s − 9-s + 2·11-s + 2i·13-s + (−1 + 2i)15-s + 6i·17-s − 8·19-s + 2·21-s − 4i·23-s + (3 + 4i)25-s + i·27-s + 8·29-s − 2i·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.894 − 0.447i)5-s + 0.755i·7-s − 0.333·9-s + 0.603·11-s + 0.554i·13-s + (−0.258 + 0.516i)15-s + 1.45i·17-s − 1.83·19-s + 0.436·21-s − 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192i·27-s + 1.48·29-s − 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.820510 + 0.507103i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.820510 + 0.507103i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2 + i)T \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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
−10.26870671570928651544769329884, −8.938326430419058969539559281165, −8.524708963394712764702786462144, −7.88366258694318355537920330272, −6.51494980513937177174276862019, −6.25788656631927173474389913661, −4.73018368441145937045797029115, −4.04529255878406932176264212297, −2.66281176407388149517619154979, −1.38493726723817759877125434325,
0.47867887922774400161696325136, 2.58222617141465845178748427393, 3.75625188599162295958960767058, 4.29048960202738611119589370171, 5.40303945019765498410426269798, 6.69776241199813181284086333673, 7.25810532710205171025243442797, 8.263121450869900726581895515883, 9.005182915485349800402921961099, 10.08363037580864116350997436415