L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.540 − 0.841i)3-s + (0.841 − 0.540i)4-s + (0.654 − 0.755i)5-s + (−0.281 + 0.959i)6-s + (0.755 + 0.654i)7-s + (−0.654 + 0.755i)8-s + (−0.415 − 0.909i)9-s + (−0.415 + 0.909i)10-s + (−0.654 − 0.755i)11-s − i·12-s + (−0.540 + 0.841i)13-s + (−0.909 − 0.415i)14-s + (−0.281 − 0.959i)15-s + (0.415 − 0.909i)16-s + (0.959 + 0.281i)17-s + ⋯ |
L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.540 − 0.841i)3-s + (0.841 − 0.540i)4-s + (0.654 − 0.755i)5-s + (−0.281 + 0.959i)6-s + (0.755 + 0.654i)7-s + (−0.654 + 0.755i)8-s + (−0.415 − 0.909i)9-s + (−0.415 + 0.909i)10-s + (−0.654 − 0.755i)11-s − i·12-s + (−0.540 + 0.841i)13-s + (−0.909 − 0.415i)14-s + (−0.281 − 0.959i)15-s + (0.415 − 0.909i)16-s + (0.959 + 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7832647388 - 0.3640741605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7832647388 - 0.3640741605i\) |
\(L(1)\) |
\(\approx\) |
\(0.8724062558 - 0.2307338927i\) |
\(L(1)\) |
\(\approx\) |
\(0.8724062558 - 0.2307338927i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 89 | \( 1 \) |
good | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.540 - 0.841i)T \) |
| 5 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.755 + 0.654i)T \) |
| 11 | \( 1 + (-0.654 - 0.755i)T \) |
| 13 | \( 1 + (-0.540 + 0.841i)T \) |
| 17 | \( 1 + (0.959 + 0.281i)T \) |
| 19 | \( 1 + (-0.909 + 0.415i)T \) |
| 23 | \( 1 + (0.909 - 0.415i)T \) |
| 29 | \( 1 + (-0.755 - 0.654i)T \) |
| 31 | \( 1 + (0.909 + 0.415i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.540 - 0.841i)T \) |
| 43 | \( 1 + (-0.755 + 0.654i)T \) |
| 47 | \( 1 + (-0.841 + 0.540i)T \) |
| 53 | \( 1 + (-0.841 - 0.540i)T \) |
| 59 | \( 1 + (0.540 + 0.841i)T \) |
| 61 | \( 1 + (0.989 + 0.142i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
| 71 | \( 1 + (0.654 + 0.755i)T \) |
| 73 | \( 1 + (0.415 - 0.909i)T \) |
| 79 | \( 1 + (-0.415 + 0.909i)T \) |
| 83 | \( 1 + (-0.281 + 0.959i)T \) |
| 97 | \( 1 + (-0.654 + 0.755i)T \) |
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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
−30.31022566967857003349237511042, −29.75638331636278612561632197793, −28.28072242198335568144379221949, −27.36750380646756417458618873139, −26.56612522608644914496134325843, −25.70257933564481451389604072306, −24.942335524951793006563412848806, −23.10166836735379514160197731021, −21.68595523583293651862245850091, −20.91569853412031181251101969751, −20.0984155281977287977329839984, −18.84002180025377246724376616538, −17.63082057910353798037119506812, −16.899533441566101589725651582152, −15.322715378353767470270607107532, −14.60595887624910534419002817679, −13.0701508769595979945463976243, −11.1702943943639361385573111023, −10.341596284021097538779701428042, −9.646456858570780536483312739074, −8.11543924516714901748812069046, −7.16924363807255495406753711034, −5.133521344850328819266289679492, −3.28408229321167864819200028942, −2.090787036141883966917376810047,
1.404254545389082849124713107627, 2.50576069255957969189034257257, 5.3159603930604495992139735159, 6.48848293971930584134433718334, 8.08765492480206546355736025261, 8.61849149739960086130657352117, 9.83433083317842519503812875060, 11.50628231311443655904586921530, 12.63642620626927584671657130263, 14.0720054202244312007786359028, 15.046865693874670683239753245592, 16.63686697030834683750961532741, 17.4609322058762362864581308885, 18.66006931339693122553868361609, 19.20031965698901900946220408563, 20.80320321516436715598886461358, 21.209127906193883425928487937609, 23.67004908607635269338282909191, 24.3458399721870803555790764093, 25.09681968534921478262746192469, 25.989179368603373774640270595114, 27.16429411924707487166110534136, 28.41673226819460179036646953657, 29.140841225855586269316079280209, 30.09461005689675254859615222086