| L(s) = 1 | + (−0.726 + 1.57i)3-s + (−2.20 + 0.341i)5-s + (−2.06 − 1.64i)7-s + (−1.94 − 2.28i)9-s + 1.06i·11-s − 4.82·13-s + (1.06 − 3.72i)15-s − 7.89i·17-s + 4.02i·19-s + (4.09 − 2.05i)21-s + 5.69·23-s + (4.76 − 1.50i)25-s + (5.00 − 1.40i)27-s + 2.00i·29-s + 4.89i·31-s + ⋯ |
| L(s) = 1 | + (−0.419 + 0.907i)3-s + (−0.988 + 0.152i)5-s + (−0.781 − 0.623i)7-s + (−0.648 − 0.761i)9-s + 0.320i·11-s − 1.33·13-s + (0.275 − 0.961i)15-s − 1.91i·17-s + 0.922i·19-s + (0.893 − 0.448i)21-s + 1.18·23-s + (0.953 − 0.301i)25-s + (0.962 − 0.269i)27-s + 0.372i·29-s + 0.879i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7297926023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7297926023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.726 - 1.57i)T \) |
| 5 | \( 1 + (2.20 - 0.341i)T \) |
| 7 | \( 1 + (2.06 + 1.64i)T \) |
| good | 11 | \( 1 - 1.06iT - 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 7.89iT - 17T^{2} \) |
| 19 | \( 1 - 4.02iT - 19T^{2} \) |
| 23 | \( 1 - 5.69T + 23T^{2} \) |
| 29 | \( 1 - 2.00iT - 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 2.56iT - 37T^{2} \) |
| 41 | \( 1 - 5.08T + 41T^{2} \) |
| 43 | \( 1 - 6.15iT - 43T^{2} \) |
| 47 | \( 1 - 2.27iT - 47T^{2} \) |
| 53 | \( 1 + 9.84T + 53T^{2} \) |
| 59 | \( 1 - 5.87T + 59T^{2} \) |
| 61 | \( 1 + 7.02iT - 61T^{2} \) |
| 67 | \( 1 + 10.8iT - 67T^{2} \) |
| 71 | \( 1 - 0.0512iT - 71T^{2} \) |
| 73 | \( 1 + 2.86T + 73T^{2} \) |
| 79 | \( 1 + 7.00T + 79T^{2} \) |
| 83 | \( 1 + 7.59iT - 83T^{2} \) |
| 89 | \( 1 - 9.72T + 89T^{2} \) |
| 97 | \( 1 - 1.77T + 97T^{2} \) |
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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
−9.606263862929709933754365103361, −8.869202150850233654606719398091, −7.59707231293550564580626630691, −7.15903637893991497176592318367, −6.28559684438945374719400662956, −4.84827709738901793244434750120, −4.73004984930576734065766021969, −3.42310000126017591177699191798, −2.91663934416748355375128202813, −0.58408767849492035069148253235,
0.57402266403118238961923159307, 2.20469484747063527974662458480, 3.12710623455196869076529147968, 4.30547625241922493814040170724, 5.32676782459678546835769481293, 6.11656761625814207179107483842, 6.97734432792039862913348605111, 7.55086169962297430478533739051, 8.454604351695991699975637073375, 9.018058546164431978352738318123
