L(s) = 1 | + (−0.00951 − 0.999i)2-s + (−0.999 + 0.0190i)4-s + (0.761 − 0.647i)5-s + (−0.0665 − 0.997i)7-s + (0.0285 + 0.999i)8-s + (−0.654 − 0.755i)10-s + (0.820 + 0.572i)13-s + (−0.997 + 0.0760i)14-s + (0.999 − 0.0380i)16-s + (0.998 + 0.0570i)17-s + (−0.254 + 0.967i)19-s + (−0.749 + 0.662i)20-s + (−0.928 − 0.371i)23-s + (0.161 − 0.986i)25-s + (0.564 − 0.825i)26-s + ⋯ |
L(s) = 1 | + (−0.00951 − 0.999i)2-s + (−0.999 + 0.0190i)4-s + (0.761 − 0.647i)5-s + (−0.0665 − 0.997i)7-s + (0.0285 + 0.999i)8-s + (−0.654 − 0.755i)10-s + (0.820 + 0.572i)13-s + (−0.997 + 0.0760i)14-s + (0.999 − 0.0380i)16-s + (0.998 + 0.0570i)17-s + (−0.254 + 0.967i)19-s + (−0.749 + 0.662i)20-s + (−0.928 − 0.371i)23-s + (0.161 − 0.986i)25-s + (0.564 − 0.825i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.542856109 + 0.01357558222i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542856109 + 0.01357558222i\) |
\(L(1)\) |
\(\approx\) |
\(0.9322976524 - 0.5077988719i\) |
\(L(1)\) |
\(\approx\) |
\(0.9322976524 - 0.5077988719i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.00951 - 0.999i)T \) |
| 5 | \( 1 + (0.761 - 0.647i)T \) |
| 7 | \( 1 + (-0.0665 - 0.997i)T \) |
| 13 | \( 1 + (0.820 + 0.572i)T \) |
| 17 | \( 1 + (0.998 + 0.0570i)T \) |
| 19 | \( 1 + (-0.254 + 0.967i)T \) |
| 23 | \( 1 + (-0.928 - 0.371i)T \) |
| 29 | \( 1 + (0.935 + 0.353i)T \) |
| 31 | \( 1 + (-0.683 + 0.730i)T \) |
| 37 | \( 1 + (-0.921 + 0.389i)T \) |
| 41 | \( 1 + (0.217 + 0.976i)T \) |
| 43 | \( 1 + (0.235 + 0.971i)T \) |
| 47 | \( 1 + (0.398 + 0.917i)T \) |
| 53 | \( 1 + (0.466 + 0.884i)T \) |
| 59 | \( 1 + (-0.953 + 0.299i)T \) |
| 61 | \( 1 + (0.861 + 0.508i)T \) |
| 67 | \( 1 + (-0.995 - 0.0950i)T \) |
| 71 | \( 1 + (0.362 + 0.931i)T \) |
| 73 | \( 1 + (-0.985 - 0.170i)T \) |
| 79 | \( 1 + (-0.991 - 0.132i)T \) |
| 83 | \( 1 + (0.969 + 0.244i)T \) |
| 89 | \( 1 + (-0.841 + 0.540i)T \) |
| 97 | \( 1 + (-0.761 - 0.647i)T \) |
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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
−21.561988800387514816078516764137, −20.6852624632955071401029571475, −19.28153948987099405253713047749, −18.71702838529928058819046018480, −17.92847750205869431420780048843, −17.5535824818934405307929534352, −16.49410350837247648744536255305, −15.59570771907346856639065093397, −15.17106163240969203954661615334, −14.21503123356208417388423395840, −13.63629006820906380492075743101, −12.79110113681493294455497796258, −11.88420707300653302234688917380, −10.665592552622978508137392120520, −9.90714176011745535726780033373, −9.076573378212152548258685307779, −8.37690981190490421410977482307, −7.38486241821967107400845495705, −6.5069673778565570410502608984, −5.68977012766838006591995601600, −5.34749228626832706401538788054, −3.89054223407615240411958670361, −2.95641359802239599729276215687, −1.81048751624384159534175016018, −0.33019798140872878453502768894,
1.203670766183185283325917590750, 1.438003013509303002469496836410, 2.83058404075049216222034715776, 3.8757675382528261423731999111, 4.51506854346145825033828126382, 5.57841071597034161312514217234, 6.41553164833078018633268308219, 7.81313442947371782902775451453, 8.5592383025303206529185315327, 9.44146097191248925119976032005, 10.24581803863618618575859219448, 10.6643468337799189864572014995, 11.86718770014888356844018270237, 12.52850632496554076254384419539, 13.29911082095008134033508143608, 14.10483159544529090085846840589, 14.38741939192679080319564537659, 16.201076486755325831887447951416, 16.61674168314396217038754380391, 17.51698101310180359792257436447, 18.18657380353901602355881758919, 19.01234332624173698465338586778, 19.90050547974197384646319019803, 20.54289405172590831677569775277, 21.12552008403739293846061140054