| L(s) = 1 | + (1.69 − 0.348i)3-s + 5-s + (2.63 − 0.272i)7-s + (2.75 − 1.18i)9-s + 3.13i·11-s + 4.34i·13-s + (1.69 − 0.348i)15-s − 0.791·17-s + 0.768i·19-s + (4.37 − 1.37i)21-s + 6.32i·23-s + 25-s + (4.26 − 2.96i)27-s − 5.96i·29-s + 5.49i·31-s + ⋯ |
| L(s) = 1 | + (0.979 − 0.201i)3-s + 0.447·5-s + (0.994 − 0.103i)7-s + (0.919 − 0.393i)9-s + 0.944i·11-s + 1.20i·13-s + (0.438 − 0.0899i)15-s − 0.191·17-s + 0.176i·19-s + (0.953 − 0.300i)21-s + 1.31i·23-s + 0.200·25-s + (0.821 − 0.570i)27-s − 1.10i·29-s + 0.986i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.026555630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.026555630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 + 0.348i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.63 + 0.272i)T \) |
| good | 11 | \( 1 - 3.13iT - 11T^{2} \) |
| 13 | \( 1 - 4.34iT - 13T^{2} \) |
| 17 | \( 1 + 0.791T + 17T^{2} \) |
| 19 | \( 1 - 0.768iT - 19T^{2} \) |
| 23 | \( 1 - 6.32iT - 23T^{2} \) |
| 29 | \( 1 + 5.96iT - 29T^{2} \) |
| 31 | \( 1 - 5.49iT - 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 + 1.60T + 41T^{2} \) |
| 43 | \( 1 + 6.18T + 43T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 + 13.1iT - 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 + 0.227iT - 71T^{2} \) |
| 73 | \( 1 + 15.7iT - 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 6.11iT - 97T^{2} \) |
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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
−9.495896040123153288787788158406, −8.553632718782968322781618401539, −7.86669330980507732965254468095, −7.12197066306791686456394809770, −6.40450520673862979198161816293, −5.05991091194783887915656391714, −4.40771684719664742127836541356, −3.40766793636404181376109342643, −2.01407504499915973176772319099, −1.64847519360148287485758609234,
1.14619414731155295249956629021, 2.40179518841148337254619223849, 3.15025289297294564293190030441, 4.29311745142582923245212873062, 5.13711320570598569298525828317, 5.99321103164728766275119087526, 7.10514904745817882942036405617, 8.009756476524381945480112315608, 8.486740104643519848482458531940, 9.103155688514804745094690212987
