L(s) = 1 | + (−0.640 + 0.768i)2-s + (−0.179 − 0.983i)4-s + (0.398 + 0.917i)5-s + (0.988 + 0.151i)7-s + (0.870 + 0.491i)8-s + (−0.959 − 0.281i)10-s + (0.879 − 0.475i)13-s + (−0.749 + 0.662i)14-s + (−0.935 + 0.353i)16-s + (−0.516 + 0.856i)17-s + (0.0855 − 0.996i)19-s + (0.830 − 0.556i)20-s + (0.888 + 0.458i)23-s + (−0.683 + 0.730i)25-s + (−0.198 + 0.980i)26-s + ⋯ |
L(s) = 1 | + (−0.640 + 0.768i)2-s + (−0.179 − 0.983i)4-s + (0.398 + 0.917i)5-s + (0.988 + 0.151i)7-s + (0.870 + 0.491i)8-s + (−0.959 − 0.281i)10-s + (0.879 − 0.475i)13-s + (−0.749 + 0.662i)14-s + (−0.935 + 0.353i)16-s + (−0.516 + 0.856i)17-s + (0.0855 − 0.996i)19-s + (0.830 − 0.556i)20-s + (0.888 + 0.458i)23-s + (−0.683 + 0.730i)25-s + (−0.198 + 0.980i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.167397895 + 1.657022290i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167397895 + 1.657022290i\) |
\(L(1)\) |
\(\approx\) |
\(0.8927001230 + 0.5083822668i\) |
\(L(1)\) |
\(\approx\) |
\(0.8927001230 + 0.5083822668i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.640 + 0.768i)T \) |
| 5 | \( 1 + (0.398 + 0.917i)T \) |
| 7 | \( 1 + (0.988 + 0.151i)T \) |
| 13 | \( 1 + (0.879 - 0.475i)T \) |
| 17 | \( 1 + (-0.516 + 0.856i)T \) |
| 19 | \( 1 + (0.0855 - 0.996i)T \) |
| 23 | \( 1 + (0.888 + 0.458i)T \) |
| 29 | \( 1 + (0.290 - 0.956i)T \) |
| 31 | \( 1 + (0.997 + 0.0760i)T \) |
| 37 | \( 1 + (0.610 + 0.791i)T \) |
| 41 | \( 1 + (-0.272 + 0.962i)T \) |
| 43 | \( 1 + (-0.995 - 0.0950i)T \) |
| 47 | \( 1 + (-0.999 - 0.0380i)T \) |
| 53 | \( 1 + (-0.774 + 0.633i)T \) |
| 59 | \( 1 + (0.969 + 0.244i)T \) |
| 61 | \( 1 + (0.345 - 0.938i)T \) |
| 67 | \( 1 + (-0.786 + 0.618i)T \) |
| 71 | \( 1 + (0.921 + 0.389i)T \) |
| 73 | \( 1 + (-0.998 - 0.0570i)T \) |
| 79 | \( 1 + (0.953 - 0.299i)T \) |
| 83 | \( 1 + (0.710 + 0.703i)T \) |
| 89 | \( 1 + (0.654 + 0.755i)T \) |
| 97 | \( 1 + (-0.398 + 0.917i)T \) |
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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
−20.93967467071374219564446908827, −20.419588074885068444694251241698, −19.5661822746596479036266643533, −18.54217093313663553320376106267, −17.99946860721243752840767546820, −17.2759834158616498431365597692, −16.48102075422337627753670262443, −15.95008154778550991027958759706, −14.54185510749801125392957341834, −13.700308915130496094210017394238, −13.06478468675623275149010948647, −12.11128427203106026546189501361, −11.47473284171231714901525842446, −10.68625846729858491638930868356, −9.80831750650665739939379459689, −8.84762435229008611853363215125, −8.48693149281952493033723094831, −7.54702114492958787371301164335, −6.4555592038876288284972168528, −5.10438938239262675167837494218, −4.50434701554786428839048155575, −3.47898626341395532464417012472, −2.181657248969108604966259081139, −1.43175692367670010859441987732, −0.62961457753532762221228361774,
0.9684485375933882501660564950, 1.89963616055498392249843235451, 2.99936742919214104586555681781, 4.4200947525976949303230742769, 5.2965881739937986054343164400, 6.26062060997487682785713958047, 6.78413494296740186476195889618, 7.92571661755497252747167727892, 8.3989445277520069171677956770, 9.426768100115376204164592808580, 10.287001932020991411509984344570, 11.064130822354530655276711148886, 11.51610020346511636164744875456, 13.3177196050093751895079064110, 13.642155619489830816327341720360, 14.88156204719650534722277774176, 15.06093743806284755493422340290, 15.88233639934855803743958254114, 17.170897579585054687452303866235, 17.56545996104597864773814559787, 18.19215954412933105054200184384, 18.92502994316992052884735986418, 19.6810952291329801574734772731, 20.68509919897295027394646092369, 21.53096308018537683432840159035