| L(s) = 1 | + 3.46i·5-s + 1.26·7-s + 4.73i·11-s − i·13-s − 5.46·17-s + 0.732i·19-s + 4·23-s − 6.99·25-s + 2i·29-s + 6.73·31-s + 4.39i·35-s + 8.92i·37-s − 8.92·41-s − 0.535i·43-s + 6.73·47-s + ⋯ |
| L(s) = 1 | + 1.54i·5-s + 0.479·7-s + 1.42i·11-s − 0.277i·13-s − 1.32·17-s + 0.167i·19-s + 0.834·23-s − 1.39·25-s + 0.371i·29-s + 1.20·31-s + 0.742i·35-s + 1.46i·37-s − 1.39·41-s − 0.0817i·43-s + 0.981·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.336829911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.336829911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 - 4.73iT - 11T^{2} \) |
| 17 | \( 1 + 5.46T + 17T^{2} \) |
| 19 | \( 1 - 0.732iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 - 8.92iT - 37T^{2} \) |
| 41 | \( 1 + 8.92T + 41T^{2} \) |
| 43 | \( 1 + 0.535iT - 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 + 2.92iT - 53T^{2} \) |
| 59 | \( 1 - 10.1iT - 59T^{2} \) |
| 61 | \( 1 + 2.92iT - 61T^{2} \) |
| 67 | \( 1 - 0.732iT - 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 5.46T + 79T^{2} \) |
| 83 | \( 1 + 3.26iT - 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−8.796113526778944632928854321111, −8.015260763211827948140487074752, −7.10344488962249749750879296522, −6.88275490615450642198634731343, −6.08985061025075853243178585576, −4.91620738448860821009956149025, −4.37510753131415477331869164057, −3.21904758878717879894230679270, −2.52948044814699542461169622185, −1.62958995975759591194702501662,
0.39414287635747183105339635758, 1.31508394645461363202472034148, 2.42705658725803742897242150660, 3.63274396101479278127290982258, 4.53652098156386998574919146442, 5.01733897881346374666627078596, 5.85202721364985541139427874737, 6.58834216968004274248243753508, 7.63963710685794925885870181253, 8.420884619767691215392730505391
