| L(s) = 1 | + 2.56i·3-s − i·7-s − 3.56·9-s + 2.12·11-s + 2i·13-s + 2.56i·17-s + 0.561·19-s + 2.56·21-s + 5.56i·23-s − 1.43i·27-s − 7.56·29-s − 0.876·31-s + 5.43i·33-s + 11.8i·37-s − 5.12·39-s + ⋯ |
| L(s) = 1 | + 1.47i·3-s − 0.377i·7-s − 1.18·9-s + 0.640·11-s + 0.554i·13-s + 0.621i·17-s + 0.128·19-s + 0.558·21-s + 1.15i·23-s − 0.276i·27-s − 1.40·29-s − 0.157·31-s + 0.946i·33-s + 1.94i·37-s − 0.820·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.343727604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343727604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - 2.56iT - 3T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2.56iT - 17T^{2} \) |
| 19 | \( 1 - 0.561T + 19T^{2} \) |
| 23 | \( 1 - 5.56iT - 23T^{2} \) |
| 29 | \( 1 + 7.56T + 29T^{2} \) |
| 31 | \( 1 + 0.876T + 31T^{2} \) |
| 37 | \( 1 - 11.8iT - 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 - 2.43iT - 43T^{2} \) |
| 47 | \( 1 + 8.24iT - 47T^{2} \) |
| 53 | \( 1 - 7.12iT - 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 + 16.1iT - 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 + 13.6iT - 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−9.944418976467930081910006114010, −9.235279674574032993740659397780, −8.599610225979331191475004391169, −7.51693522211801359190683919870, −6.56707188979173338674783248161, −5.56485505107817768026900609829, −4.73570518297217134361075641773, −3.88451374751306410474508577850, −3.34417556426098123028691884245, −1.65312673379399869202513228298,
0.54594894991251944640608802693, 1.81790461288747954917250439259, 2.70665487370864572860153316653, 3.95350283387045037418919886907, 5.31659199757292068096333569047, 6.03608714094064893431400892804, 6.93558943310981839460308802115, 7.42618400495319331025090667767, 8.346883187096622068435806675005, 8.995734075698349192308295999326
