L(s) = 1 | + (0.415 − 0.909i)2-s + (0.755 + 0.654i)3-s + (−0.654 − 0.755i)4-s + (0.959 − 0.281i)5-s + (0.909 − 0.415i)6-s + (0.281 + 0.959i)7-s + (−0.959 + 0.281i)8-s + (0.142 + 0.989i)9-s + (0.142 − 0.989i)10-s + (−0.959 − 0.281i)11-s − i·12-s + (−0.755 − 0.654i)13-s + (0.989 + 0.142i)14-s + (0.909 + 0.415i)15-s + (−0.142 + 0.989i)16-s + (−0.415 − 0.909i)17-s + ⋯ |
L(s) = 1 | + (0.415 − 0.909i)2-s + (0.755 + 0.654i)3-s + (−0.654 − 0.755i)4-s + (0.959 − 0.281i)5-s + (0.909 − 0.415i)6-s + (0.281 + 0.959i)7-s + (−0.959 + 0.281i)8-s + (0.142 + 0.989i)9-s + (0.142 − 0.989i)10-s + (−0.959 − 0.281i)11-s − i·12-s + (−0.755 − 0.654i)13-s + (0.989 + 0.142i)14-s + (0.909 + 0.415i)15-s + (−0.142 + 0.989i)16-s + (−0.415 − 0.909i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.411435152 - 0.5098995591i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.411435152 - 0.5098995591i\) |
\(L(1)\) |
\(\approx\) |
\(1.445713982 - 0.4195262532i\) |
\(L(1)\) |
\(\approx\) |
\(1.445713982 - 0.4195262532i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 89 | \( 1 \) |
good | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.755 + 0.654i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.281 + 0.959i)T \) |
| 11 | \( 1 + (-0.959 - 0.281i)T \) |
| 13 | \( 1 + (-0.755 - 0.654i)T \) |
| 17 | \( 1 + (-0.415 - 0.909i)T \) |
| 19 | \( 1 + (0.989 - 0.142i)T \) |
| 23 | \( 1 + (-0.989 + 0.142i)T \) |
| 29 | \( 1 + (-0.281 - 0.959i)T \) |
| 31 | \( 1 + (-0.989 - 0.142i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.755 + 0.654i)T \) |
| 43 | \( 1 + (-0.281 + 0.959i)T \) |
| 47 | \( 1 + (0.654 + 0.755i)T \) |
| 53 | \( 1 + (0.654 - 0.755i)T \) |
| 59 | \( 1 + (0.755 - 0.654i)T \) |
| 61 | \( 1 + (0.540 - 0.841i)T \) |
| 67 | \( 1 + (-0.654 + 0.755i)T \) |
| 71 | \( 1 + (0.959 + 0.281i)T \) |
| 73 | \( 1 + (-0.142 + 0.989i)T \) |
| 79 | \( 1 + (0.142 - 0.989i)T \) |
| 83 | \( 1 + (0.909 - 0.415i)T \) |
| 97 | \( 1 + (-0.959 + 0.281i)T \) |
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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
−30.75142235854965365158808019748, −29.9058606215198020446871939196, −28.85715648315330690909723797487, −26.77449353515381062274018695430, −26.226380446358526765357055572830, −25.42523052686571882754913381670, −24.18035277884915411285641007924, −23.7281302646049478707525278766, −22.22381590961635849851131454044, −21.14235805244204075932679984257, −20.05949417453839225237806225540, −18.39796069191305114026087753935, −17.722266096716298185524062891452, −16.59183333471468156952737242515, −15.01332063006341367649105728319, −14.097062254300044398766399888060, −13.46558916396753157594279190673, −12.384851760131397260995257194733, −10.25780747495640310334941781538, −8.97554323058272114945333704436, −7.58324289028274243528280086045, −6.88273508505335137463314773387, −5.41277493284578893744855313750, −3.77101590867553713427928545261, −2.11462035027123270784458606154,
2.155222429791525245369835483294, 2.96079263373527314565564988319, 4.8871400565729498921582688766, 5.564863913100771609501037523517, 8.17512736305045583992895331149, 9.4353664730701720967684326888, 10.04711212470243205035310813091, 11.48281052433769157442351206638, 12.93925842703346345767777405968, 13.795937570660653431795830935750, 14.86455842787810402918861223738, 15.94829116110219762468454208074, 17.82344809783457171576685382695, 18.68729176389436774431714513980, 20.153935271724170774225529300716, 20.77771363182433579013978736122, 21.82319666157730626880612602654, 22.29789210952991059354838638788, 24.242502622397990530386237762445, 25.03667793893898866343088837223, 26.36163785128190655153606195617, 27.438095950893663879900192535732, 28.4606461377308966670655563392, 29.291485737742921053454028728000, 30.54252130359318923041406764161