L(s) = 1 | + (0.623 + 0.781i)2-s + (0.365 + 0.930i)3-s + (−0.222 + 0.974i)4-s + (0.0747 − 0.997i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + (−0.900 + 0.433i)8-s + (−0.733 + 0.680i)9-s + (0.826 − 0.563i)10-s + (−0.222 − 0.974i)11-s + (−0.988 + 0.149i)12-s + (0.826 + 0.563i)13-s + (0.365 − 0.930i)14-s + (0.955 − 0.294i)15-s + (−0.900 − 0.433i)16-s + (0.0747 + 0.997i)17-s + ⋯ |
L(s) = 1 | + (0.623 + 0.781i)2-s + (0.365 + 0.930i)3-s + (−0.222 + 0.974i)4-s + (0.0747 − 0.997i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + (−0.900 + 0.433i)8-s + (−0.733 + 0.680i)9-s + (0.826 − 0.563i)10-s + (−0.222 − 0.974i)11-s + (−0.988 + 0.149i)12-s + (0.826 + 0.563i)13-s + (0.365 − 0.930i)14-s + (0.955 − 0.294i)15-s + (−0.900 − 0.433i)16-s + (0.0747 + 0.997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8484034959 + 0.7238840254i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8484034959 + 0.7238840254i\) |
\(L(1)\) |
\(\approx\) |
\(1.112498308 + 0.6645510864i\) |
\(L(1)\) |
\(\approx\) |
\(1.112498308 + 0.6645510864i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (0.623 + 0.781i)T \) |
| 3 | \( 1 + (0.365 + 0.930i)T \) |
| 5 | \( 1 + (0.0747 - 0.997i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.222 - 0.974i)T \) |
| 13 | \( 1 + (0.826 + 0.563i)T \) |
| 17 | \( 1 + (0.0747 + 0.997i)T \) |
| 19 | \( 1 + (-0.733 - 0.680i)T \) |
| 23 | \( 1 + (0.955 + 0.294i)T \) |
| 29 | \( 1 + (0.365 - 0.930i)T \) |
| 31 | \( 1 + (-0.988 + 0.149i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.623 + 0.781i)T \) |
| 47 | \( 1 + (-0.222 + 0.974i)T \) |
| 53 | \( 1 + (0.826 - 0.563i)T \) |
| 59 | \( 1 + (-0.900 - 0.433i)T \) |
| 61 | \( 1 + (-0.988 - 0.149i)T \) |
| 67 | \( 1 + (-0.733 - 0.680i)T \) |
| 71 | \( 1 + (0.955 - 0.294i)T \) |
| 73 | \( 1 + (0.826 + 0.563i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.365 + 0.930i)T \) |
| 89 | \( 1 + (0.365 + 0.930i)T \) |
| 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
−34.39118067980674391983820444919, −33.02772553046591218557131881081, −31.45527662953620240033473411192, −30.94184039025489551732138694374, −29.79998754436902218527599131966, −28.9792454234269828454880747914, −27.62850515140048283031776945268, −25.77122819197733464618868332322, −24.94187061481132174282277021702, −23.17595903950390638284750289562, −22.69878256377497212285767668381, −21.1195183750048577971123975465, −19.8438421427573398141947167728, −18.617633029667930739361682308821, −18.14094713679934933557411899508, −15.35958072296610883228430957503, −14.402847916368514632500808378035, −13.065517530267463491510043392940, −12.123818715463291868417699726259, −10.6613005523596263382415555360, −9.08328765548757633589829903603, −7.01750528120982129188148214138, −5.73565563183579492499247620217, −3.27868348410806140491048592593, −2.16398920530801459601881800336,
3.51552786162632649791928835426, 4.610412422979667021858289768281, 6.128659999511946818588170357736, 8.16817372721036357906181020443, 9.17472698544769065718607396026, 11.039233093085265251085647564850, 13.05058458149009305807353968196, 13.84923045080689374170613676172, 15.41866174627825352520825382460, 16.455577161820967170578301065722, 17.08101458144328065794852581311, 19.5105682108041306237690429218, 20.92678237023006622577162062553, 21.58273481291060429496182689135, 23.157149554445700135315192191643, 24.10782006248384739685510918438, 25.578094084909520225220224742883, 26.36449340857793578122572297926, 27.55475190235180512449230636826, 29.02580347577660053987162167880, 30.65884992850541274654970825995, 31.892278992719177793450561360749, 32.61317426167013997818981457212, 33.25045292441879439739786706462, 34.69979799594470525653595015858