L(s) = 1 | + (−0.415 + 0.909i)3-s + (0.654 − 0.755i)5-s + (0.841 − 0.540i)7-s + (−0.654 − 0.755i)9-s + (−0.142 + 0.989i)11-s + (0.841 + 0.540i)13-s + (0.415 + 0.909i)15-s + (0.959 + 0.281i)17-s + (−0.959 + 0.281i)19-s + (0.142 + 0.989i)21-s + (−0.142 − 0.989i)25-s + (0.959 − 0.281i)27-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (−0.841 − 0.540i)33-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.909i)3-s + (0.654 − 0.755i)5-s + (0.841 − 0.540i)7-s + (−0.654 − 0.755i)9-s + (−0.142 + 0.989i)11-s + (0.841 + 0.540i)13-s + (0.415 + 0.909i)15-s + (0.959 + 0.281i)17-s + (−0.959 + 0.281i)19-s + (0.142 + 0.989i)21-s + (−0.142 − 0.989i)25-s + (0.959 − 0.281i)27-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (−0.841 − 0.540i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.005435955 + 0.2055915600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.005435955 + 0.2055915600i\) |
\(L(1)\) |
\(\approx\) |
\(1.047087498 + 0.1546184415i\) |
\(L(1)\) |
\(\approx\) |
\(1.047087498 + 0.1546184415i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.841 - 0.540i)T \) |
| 11 | \( 1 + (-0.142 + 0.989i)T \) |
| 13 | \( 1 + (0.841 + 0.540i)T \) |
| 17 | \( 1 + (0.959 + 0.281i)T \) |
| 19 | \( 1 + (-0.959 + 0.281i)T \) |
| 29 | \( 1 + (-0.959 - 0.281i)T \) |
| 31 | \( 1 + (-0.415 - 0.909i)T \) |
| 37 | \( 1 + (0.654 + 0.755i)T \) |
| 41 | \( 1 + (-0.654 + 0.755i)T \) |
| 43 | \( 1 + (0.415 - 0.909i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.841 + 0.540i)T \) |
| 59 | \( 1 + (-0.841 - 0.540i)T \) |
| 61 | \( 1 + (-0.415 - 0.909i)T \) |
| 67 | \( 1 + (-0.142 - 0.989i)T \) |
| 71 | \( 1 + (0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.959 + 0.281i)T \) |
| 79 | \( 1 + (0.841 + 0.540i)T \) |
| 83 | \( 1 + (-0.654 - 0.755i)T \) |
| 89 | \( 1 + (-0.415 + 0.909i)T \) |
| 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
−30.114337334756936261111174456483, −29.525195761427416004592636217481, −28.3001552927679430646480014522, −27.322277000111060743238682766544, −25.80637504320337133447726512592, −25.05987701165261859576793399860, −23.99233958261194841031780964188, −23.00725085905271871957051473219, −21.84539106561301617666183647885, −20.94721717822399796434876767780, −19.20949619445933347536372007967, −18.37420916059201276387730406768, −17.72604601222612662372435475356, −16.46019294340854912247441595908, −14.80392051329888223582112266213, −13.86431333033487400513354901892, −12.7917403015948494233689408192, −11.36011009541642358648777753206, −10.67651482681225690624634978048, −8.75444113545337988971041462998, −7.615658545169940108560142724196, −6.17916622113078704049808910241, −5.42164759221013800138902665410, −3.00757482035795564036567281351, −1.58978911638050764628257844062,
1.65412593849831061740394380503, 4.01424419533416177891289945865, 4.946300892397152383439650440378, 6.17570990064800331849722659269, 8.065922387179721195991561725911, 9.366015737978902798843058602611, 10.34038743011368574458631036324, 11.50262335345300253547326002084, 12.82738888641163723580559736042, 14.213101464774952899083344197, 15.243285731434408702645426769, 16.7147100361398066736349291656, 17.140650317593591101109927633826, 18.38148012637396858301020481744, 20.34956291045463633538951609931, 20.83550297666698361079857587899, 21.70451242456062263337881384032, 23.158157395575899194919075842145, 23.87355127944901896214985557564, 25.39714138876286616795981492899, 26.20416794952510833773247765500, 27.68507143239856478376995287210, 28.04778803708141848994388302232, 29.195154229023246845245649313259, 30.38005115605624050608061189325