L(s) = 1 | + (−0.959 − 0.281i)3-s + (−0.415 − 0.909i)5-s + (−0.415 − 0.909i)7-s + (0.841 + 0.540i)9-s + (−0.415 + 0.909i)11-s + (−0.959 − 0.281i)13-s + (0.142 + 0.989i)15-s + (−0.142 + 0.989i)17-s + (0.841 + 0.540i)19-s + (0.142 + 0.989i)21-s + (−0.841 − 0.540i)23-s + (−0.654 + 0.755i)25-s + (−0.654 − 0.755i)27-s + (0.415 + 0.909i)29-s + (−0.841 + 0.540i)31-s + ⋯ |
L(s) = 1 | + (−0.959 − 0.281i)3-s + (−0.415 − 0.909i)5-s + (−0.415 − 0.909i)7-s + (0.841 + 0.540i)9-s + (−0.415 + 0.909i)11-s + (−0.959 − 0.281i)13-s + (0.142 + 0.989i)15-s + (−0.142 + 0.989i)17-s + (0.841 + 0.540i)19-s + (0.142 + 0.989i)21-s + (−0.841 − 0.540i)23-s + (−0.654 + 0.755i)25-s + (−0.654 − 0.755i)27-s + (0.415 + 0.909i)29-s + (−0.841 + 0.540i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5396970097 + 0.1207520948i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5396970097 + 0.1207520948i\) |
\(L(1)\) |
\(\approx\) |
\(0.6122047505 - 0.09842787759i\) |
\(L(1)\) |
\(\approx\) |
\(0.6122047505 - 0.09842787759i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 7 | \( 1 + (-0.415 - 0.909i)T \) |
| 11 | \( 1 + (-0.415 + 0.909i)T \) |
| 13 | \( 1 + (-0.959 - 0.281i)T \) |
| 17 | \( 1 + (-0.142 + 0.989i)T \) |
| 19 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-0.841 - 0.540i)T \) |
| 29 | \( 1 + (0.415 + 0.909i)T \) |
| 31 | \( 1 + (-0.841 + 0.540i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.959 - 0.281i)T \) |
| 43 | \( 1 + (0.415 - 0.909i)T \) |
| 47 | \( 1 + (-0.959 + 0.281i)T \) |
| 53 | \( 1 + (0.959 + 0.281i)T \) |
| 59 | \( 1 + (-0.959 + 0.281i)T \) |
| 61 | \( 1 + (-0.654 - 0.755i)T \) |
| 67 | \( 1 + (0.959 + 0.281i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.841 - 0.540i)T \) |
| 79 | \( 1 + (0.841 - 0.540i)T \) |
| 83 | \( 1 + (-0.142 + 0.989i)T \) |
| 97 | \( 1 + (0.415 + 0.909i)T \) |
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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
−22.54875488013005333181723884569, −21.73720735118955602880005201589, −21.45194124045286630989556391828, −19.94516338089935810525665687001, −19.14860697508990991506566642887, −18.25973819526434155392686750252, −17.97287160942609644272571387956, −16.6395414733320852638582115016, −15.94565775277033717739134156791, −15.407783615297268222635401578920, −14.44033463435692045676252726849, −13.40417147651540418290245966514, −12.306673561720927272787447050435, −11.49779589904468882615969692966, −11.18939570118684470450861320350, −9.85295061400572536532484299877, −9.45046203220065237252499281111, −7.93094730517476414016208073044, −7.089695730705914402770248351112, −6.13888554093298163445893619545, −5.46821253898728635089134417481, −4.39579741554196956019044997894, −3.19351909014551003076707738704, −2.38271525076166639327220480263, −0.40018841764358147138385549432,
0.88163569032974129959094186881, 1.997143097079746280808205546913, 3.72563184801015503858955997595, 4.594622414264799559254076924, 5.30205450982722585775643295611, 6.40214364585317040878058802420, 7.461126119308029437356939072978, 7.87543508201216843535065551981, 9.38523348609683632276686381648, 10.21171703100992509079882918459, 10.88111821816933413923855540960, 12.263232448163552528210809130383, 12.42582642229076777270083142395, 13.24137120167759094830135353755, 14.389612834025578316289841986621, 15.54667364611351443117262431478, 16.328798308035456973403182427724, 16.881532789713909802055725427525, 17.64052253323417337569820926301, 18.40223132614471557417384387962, 19.78187828700654380244822148148, 19.90645956867748913226001666788, 21.0173410272459333760225313028, 22.04062731165205252838938008588, 22.78698978396992756765889260404