L(s) = 1 | − i·3-s + 7-s − 9-s + i·11-s + i·13-s + 17-s − i·19-s − i·21-s + i·27-s − i·29-s − 31-s + 33-s + i·37-s + 39-s − 41-s + ⋯ |
L(s) = 1 | − i·3-s + 7-s − 9-s + i·11-s + i·13-s + 17-s − i·19-s − i·21-s + i·27-s − i·29-s − 31-s + 33-s + i·37-s + 39-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.741004849 - 0.3463073961i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.741004849 - 0.3463073961i\) |
\(L(1)\) |
\(\approx\) |
\(1.169250902 - 0.2413364714i\) |
\(L(1)\) |
\(\approx\) |
\(1.169250902 - 0.2413364714i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 \) |
| 89 | \( 1 \) |
| 97 | \( 1 \) |
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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
−20.33045964435108627654918161541, −19.66867154146671119498375942677, −18.58016579153359533723444694822, −18.02338853956214495300547325923, −17.03857435337689659436401169575, −16.54572691435157544524228765954, −15.830641185077471937423211758881, −14.908841246158808849048665414787, −14.44546278175566270340118366732, −13.822362330390579418060371862385, −12.6512235665619038258985193833, −11.87120480243745701772379393969, −10.952414000591824395847789513380, −10.62287674002726735686580561454, −9.74295029809876526282500317680, −8.803516444146587304861685574410, −8.169801919321091148160564228212, −7.534347696914007719348341321982, −6.04889882309016312684168200765, −5.452178533829341308884003766382, −4.87547120854572617787042746598, −3.63135339906726101402173107616, −3.30827246018607069516930271592, −2.00695348881822200575862918754, −0.78792235225692061088831548334,
0.95374682248521726260129771078, 1.876701811228682016728757609666, 2.42920133697521020313651132969, 3.741999267228939077587473976744, 4.76077128282832059930427487452, 5.43757160392510095384426544976, 6.50922881175463624451821047068, 7.231166630703275135066635288406, 7.77866864393054927809422060475, 8.66330572749147369206377628662, 9.40451268397440967041683971899, 10.45360780500137908935307351682, 11.4182484206372897438685749488, 11.8958583296285071513118962733, 12.555904799260570804071886083879, 13.55584430278094418956050283071, 14.0594056396132342465334444187, 14.840633602668489783432220967522, 15.476867919032282881321794612237, 16.862111671523470871746736256, 17.136469189024065817640424574144, 17.99395834523548499783456725593, 18.601680188708184032756198951831, 19.19737590864294420293131579497, 20.18978406612622857783243378310