L(s) = 1 | + (2.23 − 1.62i)2-s + (0.544 + 1.67i)3-s + (1.73 − 5.35i)4-s + (2.12 + 1.54i)5-s + (3.93 + 2.85i)6-s + (−0.309 + 0.951i)7-s + (−3.09 − 9.52i)8-s + (−0.0833 + 0.0605i)9-s + 7.25·10-s + 9.91·12-s + (−1.93 + 1.40i)13-s + (0.853 + 2.62i)14-s + (−1.42 + 4.39i)15-s + (−13.2 − 9.64i)16-s + (−1.93 − 1.40i)17-s + (−0.0879 + 0.270i)18-s + ⋯ |
L(s) = 1 | + (1.57 − 1.14i)2-s + (0.314 + 0.967i)3-s + (0.869 − 2.67i)4-s + (0.950 + 0.690i)5-s + (1.60 + 1.16i)6-s + (−0.116 + 0.359i)7-s + (−1.09 − 3.36i)8-s + (−0.0277 + 0.0201i)9-s + 2.29·10-s + 2.86·12-s + (−0.535 + 0.389i)13-s + (0.228 + 0.701i)14-s + (−0.369 + 1.13i)15-s + (−3.31 − 2.41i)16-s + (−0.468 − 0.340i)17-s + (−0.0207 + 0.0638i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.13449 - 1.92865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.13449 - 1.92865i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-2.23 + 1.62i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.544 - 1.67i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-2.12 - 1.54i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (1.93 - 1.40i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.93 + 1.40i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.534 - 1.64i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 0.626T + 23T^{2} \) |
| 29 | \( 1 + (0.534 - 1.64i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.81 + 1.31i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.13 - 6.55i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (3.20 + 9.87i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 + (-1.97 - 6.07i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.48 + 5.43i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.544 - 1.67i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (8.40 + 6.10i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 + (6.54 + 4.75i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.20 - 9.87i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-12.3 + 8.96i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-10.3 - 7.50i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + (-6.75 + 4.91i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.20706782636151422680163133819, −9.758873428111936105071041342608, −8.995987401804115684521138913412, −6.94929525839162309324125023329, −6.23637096425349663080537334065, −5.28626084411492727604292622771, −4.55329307496542521459380384261, −3.55891596654382744025298522774, −2.75176517843696163225953596956, −1.83525396397204908301591983170,
1.89209099181309468516671159340, 2.95609300377199411925271132969, 4.31426171099306551144981699947, 5.11449064223870203798300330688, 5.96297738832091449921465896535, 6.74576363292872223494431309936, 7.43151125260362346369519071576, 8.191976467932221738282991516735, 9.041509694586959481447319134569, 10.35895552139532424710073879584