L(s) = 1 | + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)7-s + (−0.974 − 0.222i)8-s + (−0.781 − 0.623i)10-s + (−0.974 + 0.222i)11-s + (0.222 + 0.974i)13-s + (−0.433 + 0.900i)14-s + (−0.222 − 0.974i)16-s − i·17-s + (0.781 + 0.623i)19-s + (0.222 − 0.974i)20-s + (−0.623 − 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯ |
L(s) = 1 | + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)7-s + (−0.974 − 0.222i)8-s + (−0.781 − 0.623i)10-s + (−0.974 + 0.222i)11-s + (0.222 + 0.974i)13-s + (−0.433 + 0.900i)14-s + (−0.222 − 0.974i)16-s − i·17-s + (0.781 + 0.623i)19-s + (0.222 − 0.974i)20-s + (−0.623 − 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3542728196 + 0.8807266890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3542728196 + 0.8807266890i\) |
\(L(1)\) |
\(\approx\) |
\(0.7553745800 + 0.7030482866i\) |
\(L(1)\) |
\(\approx\) |
\(0.7553745800 + 0.7030482866i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.433 + 0.900i)T \) |
| 5 | \( 1 + (-0.900 + 0.433i)T \) |
| 7 | \( 1 + (0.623 + 0.781i)T \) |
| 11 | \( 1 + (-0.974 + 0.222i)T \) |
| 13 | \( 1 + (0.222 + 0.974i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (0.781 + 0.623i)T \) |
| 23 | \( 1 + (0.900 + 0.433i)T \) |
| 31 | \( 1 + (-0.433 - 0.900i)T \) |
| 37 | \( 1 + (0.974 + 0.222i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (0.433 - 0.900i)T \) |
| 47 | \( 1 + (0.974 - 0.222i)T \) |
| 53 | \( 1 + (0.900 - 0.433i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.781 + 0.623i)T \) |
| 67 | \( 1 + (0.222 - 0.974i)T \) |
| 71 | \( 1 + (-0.222 - 0.974i)T \) |
| 73 | \( 1 + (-0.433 + 0.900i)T \) |
| 79 | \( 1 + (-0.974 - 0.222i)T \) |
| 83 | \( 1 + (-0.623 + 0.781i)T \) |
| 89 | \( 1 + (0.433 + 0.900i)T \) |
| 97 | \( 1 + (-0.781 - 0.623i)T \) |
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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
−30.5559146149552418690614380661, −29.21143308931650183089577392818, −28.222114545118318319215730850424, −27.2934409928680765237649363120, −26.45163683359203150691191355614, −24.47866581327866012080684575341, −23.60813800656553742635940359292, −22.940808776774456325027437955543, −21.48068570754627831342105370035, −20.457044212323616829372341846323, −19.83479415710897676135723430995, −18.58145858638872266808485952558, −17.40753558669617481653298901576, −15.8045823142082642563770832819, −14.747780353804784180658261718811, −13.36170877342382247441106825292, −12.53003184622763307443503758391, −11.13083534946599178701667202623, −10.50119772771181409105468632491, −8.72469479673940643525755678228, −7.57842496795840031996779403808, −5.42643307801173058124215774741, −4.34122125374951387539342939255, −3.06009331908932082598530103317, −0.982978111281199261678279721400,
2.848667571574123231337718239800, 4.40013847907254126495862282986, 5.565937861276482788966325519870, 7.15414324291779301247650582347, 7.999215966436878882566264230806, 9.284557523354463001135296268, 11.30755405435786989461598581007, 12.19077511250837601108800440245, 13.65729432296885366032889237918, 14.82179835625176955632391453725, 15.59976323226412685677486134995, 16.568641092988247143135442200585, 18.22741400886533025441298701494, 18.678722310701522983815328721061, 20.58863458548226820049009144670, 21.63493043932739930777472592838, 22.79768433191037581555066619095, 23.627980800399616680911214943790, 24.55275137374141341570842157320, 25.69134435347347885529358425003, 26.762282525600077721482367613, 27.50411794465355672853204970652, 28.884428474762510969817586230222, 30.52871386700763659797827243470, 31.31979488357886275912369330146