| L(s) = 1 | + i·3-s − 3.41i·5-s + 1.41·7-s − 9-s + 4.82i·11-s − 0.828i·13-s + 3.41·15-s + 4.82·17-s + 2.82i·19-s + 1.41i·21-s − 1.17·23-s − 6.65·25-s − i·27-s + 7.41i·29-s + 7.07·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.52i·5-s + 0.534·7-s − 0.333·9-s + 1.45i·11-s − 0.229i·13-s + 0.881·15-s + 1.17·17-s + 0.648i·19-s + 0.308i·21-s − 0.244·23-s − 1.33·25-s − 0.192i·27-s + 1.37i·29-s + 1.27·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.851650704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.851650704\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 + 3.41iT - 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 4.82iT - 11T^{2} \) |
| 13 | \( 1 + 0.828iT - 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 - 7.41iT - 29T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 + 11.6iT - 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 6.82iT - 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 1.75iT - 53T^{2} \) |
| 59 | \( 1 + 1.65iT - 59T^{2} \) |
| 61 | \( 1 - 0.343iT - 61T^{2} \) |
| 67 | \( 1 + 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 17.4T + 79T^{2} \) |
| 83 | \( 1 + 8.82iT - 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
| 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
−9.422361560508224599468130206694, −8.768021977938976622211674722813, −7.942327186063656874885192115821, −7.31178115425513598536167729455, −5.82839817279760611905709151323, −5.18218450219201128083370969165, −4.52390459294648897482734798405, −3.75428818299688049187412982155, −2.15997493957355491526568976979, −1.02083590968593160180627277632,
1.00688148998046423973535354506, 2.58708227562947187906135834894, 3.09503193862240682454721253132, 4.28880815587640424872094166506, 5.69256996572227893868721698323, 6.23428632156070182196063512697, 7.00525879278427592204392860888, 7.915337892010674828851189332264, 8.311849590901972414916399424558, 9.574890922532682282061293267054
