| 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
−9.851680752406262520431230906228, −9.401820927859221862367795635830, −8.304411787877344437373555738100, −7.21457866608892736746128228780, −6.62503022034729893819952559358, −5.80159433565197996547213704198, −4.86012450451992630628502728538, −3.94806290830205367877875884495, −3.37408654297291112437919892385, −1.67163702372311052267542055782,
0.10379604288669846574586553312, 1.15103680969589369572104194707, 3.01633603573066417763217536679, 3.42221530349621730084663114080, 5.20459636575916125006799626397, 5.45182002295232870441034582088, 6.56735350583058846380883257699, 7.11041523105580111810965995783, 8.106155826242798107741504156131, 8.571161610022939733862412314188
