L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (−0.623 + 0.781i)6-s + (−0.974 + 0.222i)7-s + (0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.433 + 0.900i)11-s + 12-s + (−0.433 + 0.900i)13-s + (0.781 + 0.623i)14-s + (−0.900 − 0.433i)16-s − 17-s + (0.900 + 0.433i)18-s + (−0.974 − 0.222i)19-s + ⋯ |
L(s) = 1 | + (−0.623 − 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (−0.623 + 0.781i)6-s + (−0.974 + 0.222i)7-s + (0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.433 + 0.900i)11-s + 12-s + (−0.433 + 0.900i)13-s + (0.781 + 0.623i)14-s + (−0.900 − 0.433i)16-s − 17-s + (0.900 + 0.433i)18-s + (−0.974 − 0.222i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1589454782 + 0.1099218802i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1589454782 + 0.1099218802i\) |
\(L(1)\) |
\(\approx\) |
\(0.4324809639 - 0.1604872038i\) |
\(L(1)\) |
\(\approx\) |
\(0.4324809639 - 0.1604872038i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 3 | \( 1 + (-0.222 - 0.974i)T \) |
| 7 | \( 1 + (-0.974 + 0.222i)T \) |
| 11 | \( 1 + (-0.433 + 0.900i)T \) |
| 13 | \( 1 + (-0.433 + 0.900i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (-0.974 - 0.222i)T \) |
| 23 | \( 1 + (0.781 + 0.623i)T \) |
| 31 | \( 1 + (0.781 - 0.623i)T \) |
| 37 | \( 1 + (-0.900 + 0.433i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (0.623 - 0.781i)T \) |
| 47 | \( 1 + (-0.900 - 0.433i)T \) |
| 53 | \( 1 + (-0.781 + 0.623i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.974 + 0.222i)T \) |
| 67 | \( 1 + (-0.433 - 0.900i)T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (-0.623 + 0.781i)T \) |
| 79 | \( 1 + (-0.433 - 0.900i)T \) |
| 83 | \( 1 + (-0.974 - 0.222i)T \) |
| 89 | \( 1 + (-0.781 + 0.623i)T \) |
| 97 | \( 1 + (-0.222 + 0.974i)T \) |
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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
−27.8016366174387232700281597069, −26.88290959825005549312271935941, −26.3040956933431301163275785954, −25.347471002314940755888935892557, −24.23713117543043147116126174063, −22.981304297029696230275156335634, −22.44523741513279094388748622667, −21.112650584907911192406942577866, −19.863699951087716603426672570641, −19.07855437716988118496480720132, −17.66765419135040171744681519840, −16.84235121019466835346926748450, −15.926737077398731645904205492198, −15.30107214958946505713958545508, −14.094282863163264534696501754257, −12.818857074289305236707430940628, −10.90380169035086133050843111217, −10.347310047818689159251842006358, −9.19514304108715113732979309118, −8.29764767196920007944914862658, −6.69690999195764074238559838168, −5.75062698486340898021406033086, −4.54403696510168022659960631867, −2.963954449447317989733951602101, −0.19969622341558657549133531057,
1.84231011182217444801707310933, 2.82348555370001283069310945859, 4.58579598509946250967567663104, 6.50820677609855050656331686432, 7.32061565557999557704962424841, 8.68032259048481324527618109278, 9.67562763752738404313170805760, 10.96100088923093423921061607289, 12.04859279718363367025964906549, 12.844739434358901662509501037272, 13.61456209647808470108049117818, 15.409504254926783831837476702871, 16.82175656986289179425749127002, 17.52569008235666729981026359519, 18.65148213467837883701215229229, 19.31523411870386245190792482770, 20.10244037715561912505396105396, 21.43668819803530204682146269919, 22.49819357134417465636860126072, 23.31827377601770501225613014146, 24.67147603697175241815776818247, 25.725060619318741640636318786, 26.29057510303633547631273784073, 27.786800009384928931833705397178, 28.72925212052469117069367118145