| L(s) = 1 | + (−1 − 1.41i)3-s − 2.82i·5-s + 2.82i·7-s + (−1.00 + 2.82i)9-s − 2·11-s − 4·13-s + (−4.00 + 2.82i)15-s − 5.65i·17-s + 2.82i·19-s + (4.00 − 2.82i)21-s − 8·23-s − 3.00·25-s + (5.00 − 1.41i)27-s + 2.82i·29-s + 8.48i·31-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.816i)3-s − 1.26i·5-s + 1.06i·7-s + (−0.333 + 0.942i)9-s − 0.603·11-s − 1.10·13-s + (−1.03 + 0.730i)15-s − 1.37i·17-s + 0.648i·19-s + (0.872 − 0.617i)21-s − 1.66·23-s − 0.600·25-s + (0.962 − 0.272i)27-s + 0.525i·29-s + 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 2.82iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.48iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.659051105619567305539615711159, −8.817817763349828461232141854330, −8.035325174956617404196389611723, −7.26106086500044222593700697668, −6.04343635614519413553257073629, −5.23119068022798486262935443023, −4.76426032514094255915795941607, −2.76486777356752179529655246344, −1.65790914525743792045177992152, 0,
2.41658397634257895216184249219, 3.68462379949509315687904134180, 4.37038468063477240948010298013, 5.62681353567438554857891473490, 6.49603968968059295844380445251, 7.31205756171170588687439624384, 8.173980253749413117220273268473, 9.644420479470472449187439539611, 10.22613264274073569266153859982
