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
−30.38005115605624050608061189325, −29.195154229023246845245649313259, −28.04778803708141848994388302232, −27.68507143239856478376995287210, −26.20416794952510833773247765500, −25.39714138876286616795981492899, −23.87355127944901896214985557564, −23.158157395575899194919075842145, −21.70451242456062263337881384032, −20.83550297666698361079857587899, −20.34956291045463633538951609931, −18.38148012637396858301020481744, −17.140650317593591101109927633826, −16.7147100361398066736349291656, −15.243285731434408702645426769, −14.213101464774952899083344197, −12.82738888641163723580559736042, −11.50262335345300253547326002084, −10.34038743011368574458631036324, −9.366015737978902798843058602611, −8.065922387179721195991561725911, −6.17570990064800331849722659269, −4.946300892397152383439650440378, −4.01424419533416177891289945865, −1.65412593849831061740394380503,
1.58978911638050764628257844062, 3.00757482035795564036567281351, 5.42164759221013800138902665410, 6.17916622113078704049808910241, 7.615658545169940108560142724196, 8.75444113545337988971041462998, 10.67651482681225690624634978048, 11.36011009541642358648777753206, 12.7917403015948494233689408192, 13.86431333033487400513354901892, 14.80392051329888223582112266213, 16.46019294340854912247441595908, 17.72604601222612662372435475356, 18.37420916059201276387730406768, 19.20949619445933347536372007967, 20.94721717822399796434876767780, 21.84539106561301617666183647885, 23.00725085905271871957051473219, 23.99233958261194841031780964188, 25.05987701165261859576793399860, 25.80637504320337133447726512592, 27.322277000111060743238682766544, 28.3001552927679430646480014522, 29.525195761427416004592636217481, 30.114337334756936261111174456483