L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.382 − 0.923i)3-s + i·4-s + (−0.923 + 0.382i)6-s + (0.923 − 0.382i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)11-s + (0.923 + 0.382i)12-s − 13-s + (−0.923 − 0.382i)14-s − 16-s + i·18-s + (−0.707 + 0.707i)19-s − i·21-s + (−0.923 − 0.382i)22-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.382 − 0.923i)3-s + i·4-s + (−0.923 + 0.382i)6-s + (0.923 − 0.382i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)11-s + (0.923 + 0.382i)12-s − 13-s + (−0.923 − 0.382i)14-s − 16-s + i·18-s + (−0.707 + 0.707i)19-s − i·21-s + (−0.923 − 0.382i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4835059596 - 0.6560763378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4835059596 - 0.6560763378i\) |
\(L(1)\) |
\(\approx\) |
\(0.7013298420 - 0.5153752784i\) |
\(L(1)\) |
\(\approx\) |
\(0.7013298420 - 0.5153752784i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.382 - 0.923i)T \) |
| 7 | \( 1 + (0.923 - 0.382i)T \) |
| 11 | \( 1 + (0.923 - 0.382i)T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 + (-0.707 + 0.707i)T \) |
| 23 | \( 1 + (-0.382 - 0.923i)T \) |
| 29 | \( 1 + (0.382 - 0.923i)T \) |
| 31 | \( 1 + (0.923 + 0.382i)T \) |
| 37 | \( 1 + (-0.382 + 0.923i)T \) |
| 41 | \( 1 + (0.382 + 0.923i)T \) |
| 43 | \( 1 + (-0.707 + 0.707i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.707 + 0.707i)T \) |
| 59 | \( 1 + (0.707 + 0.707i)T \) |
| 61 | \( 1 + (-0.382 - 0.923i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (-0.923 - 0.382i)T \) |
| 73 | \( 1 + (0.923 + 0.382i)T \) |
| 79 | \( 1 + (-0.923 + 0.382i)T \) |
| 83 | \( 1 + (-0.707 - 0.707i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (0.923 + 0.382i)T \) |
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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
−31.439642989659104765597170602747, −29.9999693058838348746613784287, −28.4022438411855266699843966525, −27.5937646718533655475754570701, −27.00627204165120269058427586556, −25.80945417405578065160080724501, −24.95534323233665975595164076419, −23.940403248702603961205365719388, −22.46938509178077259423553198483, −21.41271200446541826353570696223, −20.07325886174040779399602274119, −19.285557309310262857466432685792, −17.65563355641682347267423911003, −17.002995512539860710986811831, −15.613283443985138772569646687366, −14.82800556849841401731644579653, −14.02982498880981606069057375160, −11.75729254754939402760649703826, −10.52572455438990606368119363759, −9.36563237870010346833819800113, −8.510912612751181666942814542472, −7.199270374118139874358276351620, −5.458990980399812362231878925442, −4.38966475504819249005653061370, −2.1155700899395283929239667386,
1.24294292325636769259470645234, 2.54966463234721573519823738463, 4.23696427884595079105771809307, 6.58318633413908131986664177376, 7.85100219543112826691617175627, 8.651038553636665302868709556093, 10.124050870248236340051526870883, 11.57262720307467449073060142976, 12.307758943115994055210283106955, 13.70336601173702337621071676005, 14.71249292873629549827927551464, 16.84486504699872211126419204871, 17.507846827989395203775661419393, 18.66915527010179125297751149865, 19.560195757485165615218198956725, 20.43890626938582189633499939135, 21.55021157288570519892875858065, 22.93773321246940343105155564654, 24.42140524154161508566958267263, 25.05434420036120617704310278252, 26.47151892910547702551463427197, 27.20892570631402607720163995901, 28.43042801628379852547250954308, 29.68499102492270388028005100563, 30.10930920992300185714454190656