L(s) = 1 | + (−0.755 + 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.909 − 0.415i)13-s + (−0.540 + 0.841i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (0.540 + 0.841i)27-s + (0.841 + 0.540i)29-s + (−0.654 + 0.755i)31-s + (0.909 − 0.415i)33-s + (0.989 + 0.142i)37-s + (−0.415 + 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯ |
L(s) = 1 | + (−0.755 + 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.909 − 0.415i)13-s + (−0.540 + 0.841i)17-s + (−0.841 + 0.540i)19-s + (−0.959 + 0.281i)21-s + (0.540 + 0.841i)27-s + (0.841 + 0.540i)29-s + (−0.654 + 0.755i)31-s + (0.909 − 0.415i)33-s + (0.989 + 0.142i)37-s + (−0.415 + 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0313 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7263069040 + 0.7494260914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7263069040 + 0.7494260914i\) |
\(L(1)\) |
\(\approx\) |
\(0.8318551517 + 0.2786165971i\) |
\(L(1)\) |
\(\approx\) |
\(0.8318551517 + 0.2786165971i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 7 | \( 1 + (0.909 + 0.415i)T \) |
| 11 | \( 1 + (-0.959 - 0.281i)T \) |
| 13 | \( 1 + (0.909 - 0.415i)T \) |
| 17 | \( 1 + (-0.540 + 0.841i)T \) |
| 19 | \( 1 + (-0.841 + 0.540i)T \) |
| 29 | \( 1 + (0.841 + 0.540i)T \) |
| 31 | \( 1 + (-0.654 + 0.755i)T \) |
| 37 | \( 1 + (0.989 + 0.142i)T \) |
| 41 | \( 1 + (-0.142 - 0.989i)T \) |
| 43 | \( 1 + (0.755 - 0.654i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.909 + 0.415i)T \) |
| 59 | \( 1 + (0.415 + 0.909i)T \) |
| 61 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (0.281 + 0.959i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (0.540 + 0.841i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (-0.989 - 0.142i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (0.989 - 0.142i)T \) |
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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
−21.60674750958920460837049228171, −20.983801241409481003336392941, −20.1328636532521912191516766423, −19.171346055879033059578066541192, −18.17481233645321746302873044504, −17.984867623430294054475296130159, −17.03900737603827115375534228649, −16.23096500573511373584114143928, −15.438720349063033484263366760989, −14.35183398983689358802999913503, −13.432028803385889086266426639872, −13.00902557405355643522241666781, −11.864360635836954060816543747100, −11.08736936479983371283277955324, −10.73854213334970523255866846902, −9.47806477319065353684232879379, −8.25802229793100165660529568769, −7.68584394405253962824350313407, −6.749078845357353978899688854493, −5.94062691335456995777186305349, −4.84990548689736457221618284973, −4.33683617175668745243710726659, −2.63072911824483320500776133611, −1.76169190291850415924730313460, −0.57992382205922620357513001302,
1.13248173882581627791225831256, 2.40017602061655532920374251265, 3.69228606246914783296841265082, 4.50530858071730198371406060529, 5.520987967154757352131439415723, 5.95214984981134260127517264055, 7.169770802113805513000739588058, 8.47137835593368841804950099284, 8.74458895418532667759853597308, 10.339046813858076793361276071857, 10.62254904723327017907508336487, 11.416070122009969104011773167363, 12.36199877317444928543502367690, 13.1000099123172005684164357388, 14.25819069604466947179288234498, 15.15483792914105861174948956649, 15.655281538740817017599454554388, 16.51157432341738026247455421976, 17.37611818346633869138712778651, 18.10640548853432168661934866472, 18.58878130094980126198697393066, 19.88309077179542182417435517177, 20.8190897303367315585093354589, 21.33495066984050153118359413610, 21.87916233789430477971700428873