L(s) = 1 | + (1.34 + 0.447i)2-s − i·3-s + (1.59 + 1.20i)4-s + 1.69i·5-s + (0.447 − 1.34i)6-s + 2.41·7-s + (1.60 + 2.32i)8-s − 9-s + (−0.761 + 2.27i)10-s − 0.133i·11-s + (1.20 − 1.59i)12-s + 0.529i·13-s + (3.23 + 1.08i)14-s + 1.69·15-s + (1.11 + 3.84i)16-s − 3.72·17-s + ⋯ |
L(s) = 1 | + (0.948 + 0.316i)2-s − 0.577i·3-s + (0.799 + 0.600i)4-s + 0.760i·5-s + (0.182 − 0.547i)6-s + 0.912·7-s + (0.567 + 0.823i)8-s − 0.333·9-s + (−0.240 + 0.720i)10-s − 0.0401i·11-s + (0.346 − 0.461i)12-s + 0.146i·13-s + (0.865 + 0.289i)14-s + 0.438·15-s + (0.278 + 0.960i)16-s − 0.903·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.62006 + 0.816209i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.62006 + 0.816209i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.34 - 0.447i)T \) |
| 3 | \( 1 + iT \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 1.69iT - 5T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 + 0.133iT - 11T^{2} \) |
| 13 | \( 1 - 0.529iT - 13T^{2} \) |
| 17 | \( 1 + 3.72T + 17T^{2} \) |
| 19 | \( 1 + 1.26iT - 19T^{2} \) |
| 29 | \( 1 + 2.58iT - 29T^{2} \) |
| 31 | \( 1 - 4.37T + 31T^{2} \) |
| 37 | \( 1 + 3.36iT - 37T^{2} \) |
| 41 | \( 1 + 5.68T + 41T^{2} \) |
| 43 | \( 1 + 2.62iT - 43T^{2} \) |
| 47 | \( 1 + 0.552T + 47T^{2} \) |
| 53 | \( 1 + 10.6iT - 53T^{2} \) |
| 59 | \( 1 - 3.59iT - 59T^{2} \) |
| 61 | \( 1 + 6.75iT - 61T^{2} \) |
| 67 | \( 1 + 0.405iT - 67T^{2} \) |
| 71 | \( 1 + 5.62T + 71T^{2} \) |
| 73 | \( 1 + 8.66T + 73T^{2} \) |
| 79 | \( 1 + 2.77T + 79T^{2} \) |
| 83 | \( 1 + 1.80iT - 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 5.11T + 97T^{2} \) |
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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
−11.25864403495544905642933358074, −10.34483765008283238607117306462, −8.761443075482908886190557926936, −7.945682096509684529061250450700, −7.02132475440379461514499997676, −6.44697531642386465719510124957, −5.29116071034074319520293946189, −4.32718839269079236140179512390, −2.99076484094765473824418284237, −1.93535474552577589910394729944,
1.45756527664589337612223891607, 2.92185226955343166934717621606, 4.32265398934825406336614454623, 4.81042031123102470920792414163, 5.71690827076459059227866364019, 6.88690804330639714343386267312, 8.129400470169362061524314756809, 8.978769703508626920294101085879, 10.06688577652505306711356044890, 10.87555420438530620297995931330