L(s) = 1 | + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.654 + 0.755i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.415 − 0.909i)12-s + (0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (0.654 − 0.755i)17-s + (−0.841 + 0.540i)18-s + (−0.654 − 0.755i)19-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.654 + 0.755i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.415 − 0.909i)12-s + (0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (0.654 − 0.755i)17-s + (−0.841 + 0.540i)18-s + (−0.654 − 0.755i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9653060489 + 0.5931053764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9653060489 + 0.5931053764i\) |
\(L(1)\) |
\(\approx\) |
\(1.017174988 + 0.4563193137i\) |
\(L(1)\) |
\(\approx\) |
\(1.017174988 + 0.4563193137i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 11 | \( 1 + (0.415 + 0.909i)T \) |
| 13 | \( 1 + (0.142 + 0.989i)T \) |
| 17 | \( 1 + (0.654 - 0.755i)T \) |
| 19 | \( 1 + (-0.654 - 0.755i)T \) |
| 29 | \( 1 + (-0.654 + 0.755i)T \) |
| 31 | \( 1 + (-0.959 + 0.281i)T \) |
| 37 | \( 1 + (-0.841 - 0.540i)T \) |
| 41 | \( 1 + (0.841 - 0.540i)T \) |
| 43 | \( 1 + (0.959 + 0.281i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.142 - 0.989i)T \) |
| 59 | \( 1 + (-0.142 - 0.989i)T \) |
| 61 | \( 1 + (-0.959 + 0.281i)T \) |
| 67 | \( 1 + (-0.415 + 0.909i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.654 + 0.755i)T \) |
| 79 | \( 1 + (-0.142 - 0.989i)T \) |
| 83 | \( 1 + (-0.841 - 0.540i)T \) |
| 89 | \( 1 + (-0.959 - 0.281i)T \) |
| 97 | \( 1 + (-0.841 + 0.540i)T \) |
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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
−29.36168942459834257453357744674, −27.89522671498788679021518349605, −27.32964855871603707270038921905, −26.07039660679023265196630762666, −25.28995058231709820058737083207, −24.27196549210104892262650497322, −22.6857169719161010775486708084, −21.50671504891928325535223493490, −20.888972981284871630264652893789, −19.66725898635873456311540660030, −18.92182229145737689738769900442, −18.17237261214156329209068104310, −16.80619228902294691006193847693, −15.27451433761165154309998100791, −14.198640256400056312899002298075, −12.96451213138219517006268449095, −12.208983560931118953909044853384, −10.82394716165094679768161103585, −9.55335524912282323295813576583, −8.54050142197155784609331305883, −7.85808135720847097829769933129, −5.83995369832252540026800815709, −3.868789648886395736836210209041, −2.84825592731097388141068502234, −1.54669719908699337743019671743,
1.70032990617069532181482732322, 3.888825212390810721720376545056, 4.863289122144121464710192537603, 6.8969248661648799433229533012, 7.49960292134606028283969422870, 8.918876567880722626651802785689, 9.67859129120499980244253771669, 10.86280348254963788810770601946, 12.93747337003145703831928385795, 14.134629719278438832158581938521, 14.60768289402334270004947763097, 15.915777624847398639744964324675, 16.78510924628641054327458507918, 17.95110756570439161392318494907, 19.18619292369191165187778143224, 19.987497740257195507372891708942, 21.08234902101288502013531782401, 22.53347374958893920811895992631, 23.63265708304854200554671984976, 24.52018410703674215696434248513, 25.7608203135735083326283690359, 26.102170171597545056498568860772, 27.27461167053336546402224654739, 27.92335739415951664164956648931, 29.49924123700595537978066449112