L(s) = 1 | + (0.415 − 0.909i)3-s + (0.142 − 0.989i)5-s + (0.142 − 0.989i)7-s + (−0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.415 + 0.909i)13-s + (−0.841 − 0.540i)15-s + (0.841 − 0.540i)17-s + (−0.654 − 0.755i)19-s + (−0.841 − 0.540i)21-s + (0.654 + 0.755i)23-s + (−0.959 − 0.281i)25-s + (−0.959 + 0.281i)27-s + (0.142 − 0.989i)29-s + (0.654 − 0.755i)31-s + ⋯ |
L(s) = 1 | + (0.415 − 0.909i)3-s + (0.142 − 0.989i)5-s + (0.142 − 0.989i)7-s + (−0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.415 + 0.909i)13-s + (−0.841 − 0.540i)15-s + (0.841 − 0.540i)17-s + (−0.654 − 0.755i)19-s + (−0.841 − 0.540i)21-s + (0.654 + 0.755i)23-s + (−0.959 − 0.281i)25-s + (−0.959 + 0.281i)27-s + (0.142 − 0.989i)29-s + (0.654 − 0.755i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.6491988095 - 1.662417565i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.6491988095 - 1.662417565i\) |
\(L(1)\) |
\(\approx\) |
\(0.8039737765 - 0.8361792549i\) |
\(L(1)\) |
\(\approx\) |
\(0.8039737765 - 0.8361792549i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 11 | \( 1 + (-0.142 - 0.989i)T \) |
| 13 | \( 1 + (-0.415 + 0.909i)T \) |
| 17 | \( 1 + (0.841 - 0.540i)T \) |
| 19 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (0.654 + 0.755i)T \) |
| 29 | \( 1 + (0.142 - 0.989i)T \) |
| 31 | \( 1 + (0.654 - 0.755i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.415 + 0.909i)T \) |
| 43 | \( 1 + (-0.142 - 0.989i)T \) |
| 47 | \( 1 + (-0.415 - 0.909i)T \) |
| 53 | \( 1 + (-0.415 + 0.909i)T \) |
| 59 | \( 1 + (0.415 + 0.909i)T \) |
| 61 | \( 1 + (0.959 - 0.281i)T \) |
| 67 | \( 1 + (0.415 - 0.909i)T \) |
| 71 | \( 1 + (0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.654 + 0.755i)T \) |
| 79 | \( 1 + (0.654 - 0.755i)T \) |
| 83 | \( 1 + (0.841 - 0.540i)T \) |
| 97 | \( 1 + (-0.142 + 0.989i)T \) |
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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
−22.54069437087653809702737257777, −22.24558294424043792232494061526, −21.1111670769970777919829043159, −20.83214814939621397462475995644, −19.51873666071992892178963650637, −19.0061706244837288408571769664, −17.94969442661474519938129270002, −17.31013270160071962277565693440, −16.11101596773182672023445462436, −15.3287234838474939967046742773, −14.62694322411760971391266769286, −14.413646789843078340318035208639, −12.83727929695208195542738298757, −12.16225340213057881369753541030, −10.90827880000840966268182284877, −10.29386101905539777144684427033, −9.65809256505920148703300711304, −8.53639382409222768082522430162, −7.815786763943127402120238427390, −6.629162818480254342558727628049, −5.55050009756183294718801449298, −4.815260208989143426317080272191, −3.51367621128587109098075312894, −2.784110153343169722221494295752, −1.90129234351190665312483827739,
0.40516594598534980250706593618, 1.08886558660410433083327041725, 2.208473229834739340039095721013, 3.435889926381460933537083683035, 4.4837846590330084544579499724, 5.5431132381704080420149744376, 6.60841761197721033870397361633, 7.47514806265031731087548057298, 8.27197035561694996444303336357, 9.07101546458740332580129463865, 9.938918318893035716620552764692, 11.31998786632447805754086177304, 11.90391402154791190067356080870, 12.99744070945524523947732447121, 13.62812922196012710842416780291, 14.05920964748187438801295706447, 15.25424835426171473958045126978, 16.44648432408050109621040665186, 17.03863952051188257626279229918, 17.64610235425738269691249289775, 19.02209330870497230955823768767, 19.2658639087298277849097951966, 20.27161069700834380027707776643, 20.951806044596987552299731102424, 21.62299385919050688409467898176