| L(s) = 1 | + (1.10 − 1.33i)3-s − 5-s + 2.36i·7-s + (−0.540 − 2.95i)9-s + 0.927·11-s + 1.43i·13-s + (−1.10 + 1.33i)15-s − 5.72i·17-s − 1.85·19-s + (3.14 + 2.62i)21-s + 7.68i·23-s + 25-s + (−4.52 − 2.55i)27-s − 7.94i·29-s − 0.732i·31-s + ⋯ |
| L(s) = 1 | + (0.640 − 0.768i)3-s − 0.447·5-s + 0.893i·7-s + (−0.180 − 0.983i)9-s + 0.279·11-s + 0.397i·13-s + (−0.286 + 0.343i)15-s − 1.38i·17-s − 0.426·19-s + (0.686 + 0.572i)21-s + 1.60i·23-s + 0.200·25-s + (−0.870 − 0.491i)27-s − 1.47i·29-s − 0.131i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.794716953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.794716953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.10 + 1.33i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (7.02 - 4.19i)T \) |
| good | 7 | \( 1 - 2.36iT - 7T^{2} \) |
| 11 | \( 1 - 0.927T + 11T^{2} \) |
| 13 | \( 1 - 1.43iT - 13T^{2} \) |
| 17 | \( 1 + 5.72iT - 17T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 - 7.68iT - 23T^{2} \) |
| 29 | \( 1 + 7.94iT - 29T^{2} \) |
| 31 | \( 1 + 0.732iT - 31T^{2} \) |
| 37 | \( 1 + 1.54T + 37T^{2} \) |
| 41 | \( 1 - 6.32T + 41T^{2} \) |
| 43 | \( 1 + 5.59iT - 43T^{2} \) |
| 47 | \( 1 + 9.43iT - 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 6.47iT - 59T^{2} \) |
| 61 | \( 1 + 13.3iT - 61T^{2} \) |
| 71 | \( 1 + 5.24iT - 71T^{2} \) |
| 73 | \( 1 + 3.58T + 73T^{2} \) |
| 79 | \( 1 + 12.6iT - 79T^{2} \) |
| 83 | \( 1 - 4.64iT - 83T^{2} \) |
| 89 | \( 1 + 1.75iT - 89T^{2} \) |
| 97 | \( 1 + 12.2iT - 97T^{2} \) |
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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
−8.231540607582350197086618684558, −7.44735477145939413786387406293, −7.03076322086621693449586733680, −6.06048444888649749735834964919, −5.42593324803990684242469009741, −4.29733899101448136146282517157, −3.47802291452401328753685074195, −2.57688543967774457751549853062, −1.86520913001935312648144896216, −0.50765257801598194214151598222,
1.14616717712712335166585257217, 2.45454446118633762682353989307, 3.36885173055574863863619583659, 4.14728431037281769395163979283, 4.50977006687133953902807734426, 5.59817792892881435984666072895, 6.53006818012298650536052210654, 7.32378780255263725290712905590, 8.068111802134043104634096146217, 8.607132803987143852642894317954
