| L(s) = 1 | + (−1.31 − 1.12i)3-s − 5-s + (−2.64 + 0.0644i)7-s + (0.468 + 2.96i)9-s − 2.69i·11-s − 6.61i·13-s + (1.31 + 1.12i)15-s + 7.03·17-s − 3.22i·19-s + (3.55 + 2.89i)21-s − 2.56i·23-s + 25-s + (2.71 − 4.42i)27-s + 8.21i·29-s + 2.87i·31-s + ⋯ |
| L(s) = 1 | + (−0.760 − 0.649i)3-s − 0.447·5-s + (−0.999 + 0.0243i)7-s + (0.156 + 0.987i)9-s − 0.813i·11-s − 1.83i·13-s + (0.340 + 0.290i)15-s + 1.70·17-s − 0.740i·19-s + (0.775 + 0.630i)21-s − 0.534i·23-s + 0.200·25-s + (0.522 − 0.852i)27-s + 1.52i·29-s + 0.515i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2145680289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2145680289\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.31 + 1.12i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2.64 - 0.0644i)T \) |
| good | 11 | \( 1 + 2.69iT - 11T^{2} \) |
| 13 | \( 1 + 6.61iT - 13T^{2} \) |
| 17 | \( 1 - 7.03T + 17T^{2} \) |
| 19 | \( 1 + 3.22iT - 19T^{2} \) |
| 23 | \( 1 + 2.56iT - 23T^{2} \) |
| 29 | \( 1 - 8.21iT - 29T^{2} \) |
| 31 | \( 1 - 2.87iT - 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 2.56T + 47T^{2} \) |
| 53 | \( 1 - 2.70iT - 53T^{2} \) |
| 59 | \( 1 + 3.03T + 59T^{2} \) |
| 61 | \( 1 + 7.83iT - 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 14.0iT - 71T^{2} \) |
| 73 | \( 1 - 1.81iT - 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 3.01T + 83T^{2} \) |
| 89 | \( 1 + 4.17T + 89T^{2} \) |
| 97 | \( 1 - 9.55iT - 97T^{2} \) |
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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
−8.571161610022939733862412314188, −8.106155826242798107741504156131, −7.11041523105580111810965995783, −6.56735350583058846380883257699, −5.45182002295232870441034582088, −5.20459636575916125006799626397, −3.42221530349621730084663114080, −3.01633603573066417763217536679, −1.15103680969589369572104194707, −0.10379604288669846574586553312,
1.67163702372311052267542055782, 3.37408654297291112437919892385, 3.94806290830205367877875884495, 4.86012450451992630628502728538, 5.80159433565197996547213704198, 6.62503022034729893819952559358, 7.21457866608892736746128228780, 8.304411787877344437373555738100, 9.401820927859221862367795635830, 9.851680752406262520431230906228
