L(s) = 1 | + 5-s − 11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s − 23-s + 25-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + (−0.5 + 0.866i)53-s − 55-s + ⋯ |
L(s) = 1 | + 5-s − 11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s − 23-s + 25-s + (−0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.5 − 0.866i)47-s + (−0.5 + 0.866i)53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7232579281 + 1.105909586i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7232579281 + 1.105909586i\) |
\(L(1)\) |
\(\approx\) |
\(1.021425531 + 0.2308952016i\) |
\(L(1)\) |
\(\approx\) |
\(1.021425531 + 0.2308952016i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−25.6072014454123606846292532567, −24.53690075616478297499533327214, −23.87385268983195201134960979864, −22.39306053147840461402649030598, −22.05802828481465778736956587150, −20.70503389286487291528460498650, −20.27887862955715783887345087839, −18.82495781752433530154664678260, −17.93407616368839729110342912709, −17.34436068420998868761397942530, −16.06628627209880902375410773214, −15.23740641621182069163891757843, −13.9601412629241911980867192082, −13.31352581974860447868269976607, −12.319268610609507769599546177156, −10.97203544090098589169915445718, −10.07592531963679657941923244606, −9.21986002924554366168697482783, −7.94364439946902832493878940489, −6.85769417170000329177176316369, −5.58047045787562809269519905878, −4.87096949085885975988827686409, −3.04553766168942424031064716981, −2.10994337600446301953291861113, −0.385701533562361819059990040996,
1.61542692321552271865875450331, 2.60760509017926336389638239076, 4.19346447929159342797457650737, 5.441458815060059624921288787137, 6.307289711660895530382058658688, 7.568462044346576710328694424782, 8.733797628891699662669864620920, 9.862348935021183614564753480759, 10.52066719876684225550003368567, 11.87045686145918525775361194291, 12.92293970187875612134284147425, 13.80215387421080880947649132763, 14.641883629097987645378577276183, 15.86580620167086942601125685497, 16.80825152452723743174195429032, 17.74902039016309399582746626356, 18.52098277698656006699116278066, 19.586246436125593120412338098699, 20.76372698592695943867041370823, 21.43710428109325304620719658671, 22.23818044407413134723899284157, 23.392036445713547887764916281127, 24.334570005439574598176205187202, 25.09699320048238171698480792888, 26.32117185691257409647594743550