| L(s) = 1 | − 2.82i·3-s + 2i·7-s − 5.00·9-s − 11-s + 1.17i·13-s + 6.82i·17-s + 5.65·21-s − 2.82i·23-s + 5.65i·27-s + 3.65·29-s + 2.82i·33-s − 7.65i·37-s + 3.31·39-s + 6·41-s − 6i·43-s + ⋯ |
| L(s) = 1 | − 1.63i·3-s + 0.755i·7-s − 1.66·9-s − 0.301·11-s + 0.324i·13-s + 1.65i·17-s + 1.23·21-s − 0.589i·23-s + 1.08i·27-s + 0.679·29-s + 0.492i·33-s − 1.25i·37-s + 0.530·39-s + 0.937·41-s − 0.914i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.514170603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.514170603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.82iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 13 | \( 1 - 1.17iT - 13T^{2} \) |
| 17 | \( 1 - 6.82iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.65iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 11.6iT - 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 + 12.4iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 1.17iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 3.65iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.097088317272641348842707485994, −7.40952247945001480181320481129, −6.64594014146364659764338189534, −6.06659940746632220319929642121, −5.52223581986633535639050425356, −4.37713910163917445079960046644, −3.27576339280187320854943509543, −2.22536324560165603582955977948, −1.80676570909815978151741318167, −0.51048771559408524682035796411,
0.938972446594747938477046093092, 2.73563435579695915509853904134, 3.23792134015048374207956141859, 4.27391961715035720770679378923, 4.69121595347927554238713755187, 5.40698982628883633153081403588, 6.24547716525890975262587485357, 7.31752819117051787998564231624, 7.83674854048726025273114317847, 8.911865569284517631747626077883
