L(s) = 1 | + (−0.866 + 0.5i)2-s − 3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s + (0.866 − 0.5i)7-s + i·8-s + 9-s + (0.5 − 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)18-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s − 3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s + (0.866 − 0.5i)7-s + i·8-s + 9-s + (0.5 − 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5523654168 + 0.05718897400i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5523654168 + 0.05718897400i\) |
\(L(1)\) |
\(\approx\) |
\(0.5315564690 + 0.08974834694i\) |
\(L(1)\) |
\(\approx\) |
\(0.5315564690 + 0.08974834694i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.866 + 0.5i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.866 - 0.5i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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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
−24.28063467051295976544510776539, −23.66190575237257211938297605125, −22.384418442893795949027768820033, −21.58726587553418703506813539536, −20.96959164796431432584015384989, −19.658046641864692197947157095, −19.21718313514803346785585109922, −18.10895026483732315347476700093, −17.34097666399084174512088029967, −16.75298836831092399579039064869, −15.74594479548110680108441418154, −15.045867834677908437131725905009, −13.24587408043918044736704632176, −12.29418453154215495817900128492, −11.50827920081843144160623824025, −11.22694750063332064432111122673, −10.01989661812643834518249792100, −8.77193987916013797107753700928, −8.18517479022642103664332545181, −7.03111285904467525589675566637, −5.97284788024202056613260438928, −4.56855339755178631579363345474, −3.79293349135182489030410170762, −1.97689412748316285901164136623, −0.89424416888779005724894885547,
0.72357518784284036816999257817, 2.049119844448536838006919582168, 4.10909926939803474051404330593, 4.89790767833420165333974377770, 6.30703200601255302354698908356, 7.00906517069983881665740819573, 7.75736096676191916081028297975, 8.868921318432395220100688510630, 10.177112188894180677613162325405, 10.8833350000466220777574669528, 11.55163645374821864050042377026, 12.40344701270989866063896638191, 14.20479757945119632819775976668, 14.820518913094941255390148935493, 15.869827816847359666957033847036, 16.553750614081088455004701710032, 17.47853468462026372113404418573, 18.07135567386143042097928379895, 18.90063997716903925712764946600, 19.88356624537387601328476707817, 20.67783540491763681093844585641, 22.036083020698393318066113952766, 23.00330045782449400944786683727, 23.497888828468105478251245221596, 24.361115490377515329521566224378