| L(s) = 1 | + 1.73·3-s + (4.25 − 2.62i)5-s + (−2.56 + 6.51i)7-s + 2.99·9-s − 8.37·11-s − 0.0883·13-s + (7.37 − 4.54i)15-s − 29.7·17-s − 17.8i·19-s + (−4.44 + 11.2i)21-s + 39.2i·23-s + (11.2 − 22.3i)25-s + 5.19·27-s + 43.7·29-s + 53.8i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (0.851 − 0.524i)5-s + (−0.366 + 0.930i)7-s + 0.333·9-s − 0.761·11-s − 0.00679·13-s + (0.491 − 0.303i)15-s − 1.75·17-s − 0.941i·19-s + (−0.211 + 0.537i)21-s + 1.70i·23-s + (0.448 − 0.893i)25-s + 0.192·27-s + 1.50·29-s + 1.73i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.780546099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780546099\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 + (-4.25 + 2.62i)T \) |
| 7 | \( 1 + (2.56 - 6.51i)T \) |
| good | 11 | \( 1 + 8.37T + 121T^{2} \) |
| 13 | \( 1 + 0.0883T + 169T^{2} \) |
| 17 | \( 1 + 29.7T + 289T^{2} \) |
| 19 | \( 1 + 17.8iT - 361T^{2} \) |
| 23 | \( 1 - 39.2iT - 529T^{2} \) |
| 29 | \( 1 - 43.7T + 841T^{2} \) |
| 31 | \( 1 - 53.8iT - 961T^{2} \) |
| 37 | \( 1 - 27.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 74.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 27.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 5.38iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 1.93iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 74.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 30.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 72.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 34.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 125.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 92.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 141. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 107.T + 9.40e3T^{2} \) |
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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
−9.364777494567375551361158148952, −8.655917218706738199802682918924, −8.114203904065625113653368957924, −6.81263065605485251416267103059, −6.27898289769310743054435672632, −5.11526195899292000316228189932, −4.69713338754817188596049983489, −3.09015380160457797435420029514, −2.48992699241162127249345862408, −1.41832566416177049809667005918,
0.40272915637291146762889606104, 2.06185013806578199641926802833, 2.67250408692090538586563244759, 3.86792258219754941209239199450, 4.63616999293722299957044462820, 5.87281177524631634134357529917, 6.64593589514050937847464648551, 7.24076484860320921948772949985, 8.224158313642534093632691522584, 8.944916993016375835879740093713
