L(s) = 1 | + (0.352 + 0.935i)3-s + (−0.729 − 0.683i)5-s + (−0.751 + 0.659i)9-s + (0.582 − 0.812i)11-s + (0.290 − 0.956i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (0.849 + 0.528i)19-s + (−0.997 − 0.0654i)23-s + (0.0654 + 0.997i)25-s + (−0.881 − 0.471i)27-s + (0.0980 + 0.995i)29-s + (−0.965 + 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.999 + 0.0327i)37-s + ⋯ |
L(s) = 1 | + (0.352 + 0.935i)3-s + (−0.729 − 0.683i)5-s + (−0.751 + 0.659i)9-s + (0.582 − 0.812i)11-s + (0.290 − 0.956i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (0.849 + 0.528i)19-s + (−0.997 − 0.0654i)23-s + (0.0654 + 0.997i)25-s + (−0.881 − 0.471i)27-s + (0.0980 + 0.995i)29-s + (−0.965 + 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.999 + 0.0327i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.057964394 + 0.09052529530i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.057964394 + 0.09052529530i\) |
\(L(1)\) |
\(\approx\) |
\(1.092131483 + 0.1390622640i\) |
\(L(1)\) |
\(\approx\) |
\(1.092131483 + 0.1390622640i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.352 + 0.935i)T \) |
| 5 | \( 1 + (-0.729 - 0.683i)T \) |
| 11 | \( 1 + (0.582 - 0.812i)T \) |
| 13 | \( 1 + (0.290 - 0.956i)T \) |
| 17 | \( 1 + (0.991 + 0.130i)T \) |
| 19 | \( 1 + (0.849 + 0.528i)T \) |
| 23 | \( 1 + (-0.997 - 0.0654i)T \) |
| 29 | \( 1 + (0.0980 + 0.995i)T \) |
| 31 | \( 1 + (-0.965 + 0.258i)T \) |
| 37 | \( 1 + (-0.999 + 0.0327i)T \) |
| 41 | \( 1 + (0.831 + 0.555i)T \) |
| 43 | \( 1 + (-0.634 - 0.773i)T \) |
| 47 | \( 1 + (0.793 + 0.608i)T \) |
| 53 | \( 1 + (0.812 + 0.582i)T \) |
| 59 | \( 1 + (0.973 + 0.227i)T \) |
| 61 | \( 1 + (0.162 - 0.986i)T \) |
| 67 | \( 1 + (-0.935 + 0.352i)T \) |
| 71 | \( 1 + (-0.195 + 0.980i)T \) |
| 73 | \( 1 + (0.321 - 0.946i)T \) |
| 79 | \( 1 + (-0.130 - 0.991i)T \) |
| 83 | \( 1 + (-0.471 - 0.881i)T \) |
| 89 | \( 1 + (-0.442 + 0.896i)T \) |
| 97 | \( 1 + (0.707 + 0.707i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.73245110526303508959587037129, −19.38485115602494575873847216454, −18.47921713486891062539739485656, −18.11195908478737915724068709485, −17.20393130320950193780233527817, −16.30935672926218617721695300831, −15.46901970587906034765787885972, −14.61901944557344160166401362916, −14.1718239584719619523751328978, −13.44353258860423669609395465798, −12.35146574181763523468106881308, −11.82735004755851858228682577517, −11.37822183479792108138239983899, −10.14582682239165031491835754614, −9.36536366223010671527833821367, −8.495518168064076505742897626615, −7.57290757575124965202383396099, −7.16635425160900559314210316119, −6.43806218577969029942521905266, −5.49899121432348358187869427412, −4.13973504675844376778315372309, −3.57878095840908548924855364429, −2.52359913458481631152519233538, −1.73467675582837605160375615093, −0.652982314084997204725544984462,
0.54516882413524850198364140118, 1.55065081880333570792482378755, 3.1550005561978488146969600502, 3.5204881182617146509302673210, 4.31504050842479673379557819394, 5.45902723022105962407204580808, 5.70989798643250088453769529296, 7.29251824157117508248816148126, 8.08787337100910291294793648889, 8.62308374174762217761780116950, 9.37073503501188413329343103725, 10.26553460535844981501162713255, 10.92520044915077635486170909714, 11.83874364473455270807052179644, 12.40732343306518337706901102467, 13.47459237564248126284401456590, 14.25409494055924838098025757761, 14.858057460299395634534300585828, 15.82156241152580755258438395086, 16.23756014532591995933279923290, 16.75588968267563670494598538856, 17.7309444912412289276885372372, 18.754431349030090908917546823430, 19.47345686187989798520063738535, 20.28635739042102781234292062497