L(s) = 1 | + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)5-s + (0.623 − 0.781i)8-s + (−0.623 − 0.781i)10-s + (−0.974 − 0.222i)11-s + (−0.974 − 0.222i)13-s + (0.623 + 0.781i)16-s + (−0.433 − 0.900i)17-s − i·19-s + (0.900 − 0.433i)20-s + (0.433 − 0.900i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.433 − 0.900i)26-s + ⋯ |
L(s) = 1 | + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)5-s + (0.623 − 0.781i)8-s + (−0.623 − 0.781i)10-s + (−0.974 − 0.222i)11-s + (−0.974 − 0.222i)13-s + (0.623 + 0.781i)16-s + (−0.433 − 0.900i)17-s − i·19-s + (0.900 − 0.433i)20-s + (0.433 − 0.900i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.433 − 0.900i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04380113578 + 0.1988870438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04380113578 + 0.1988870438i\) |
\(L(1)\) |
\(\approx\) |
\(0.5185228971 + 0.2171504565i\) |
\(L(1)\) |
\(\approx\) |
\(0.5185228971 + 0.2171504565i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (-0.222 + 0.974i)T \) |
| 5 | \( 1 + (-0.623 + 0.781i)T \) |
| 11 | \( 1 + (-0.974 - 0.222i)T \) |
| 13 | \( 1 + (-0.974 - 0.222i)T \) |
| 17 | \( 1 + (-0.433 - 0.900i)T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (0.900 + 0.433i)T \) |
| 29 | \( 1 + (-0.433 - 0.900i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.900 + 0.433i)T \) |
| 43 | \( 1 + (-0.623 - 0.781i)T \) |
| 47 | \( 1 + (-0.974 - 0.222i)T \) |
| 53 | \( 1 + (0.433 - 0.900i)T \) |
| 59 | \( 1 + (0.623 + 0.781i)T \) |
| 61 | \( 1 + (-0.900 + 0.433i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.433 - 0.900i)T \) |
| 73 | \( 1 + (-0.222 - 0.974i)T \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 + (-0.222 - 0.974i)T \) |
| 89 | \( 1 + (-0.974 + 0.222i)T \) |
| 97 | \( 1 - iT \) |
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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
−17.43691003629879405988827532812, −16.80483007430346891186355640847, −16.39889763365801013380780213, −15.42588839204636013230831191406, −14.76493400128198289293557771499, −14.08712557876361349575684600225, −13.05640441925800646358214877997, −12.63363079602812978523528349777, −12.40238999797211794890585058809, −11.40118930904206617565534474481, −10.88090991867960518676988739612, −10.1906516639777425928580821830, −9.48459707909824622450092870241, −8.81225687081523741119887923092, −8.17222684061166440240479125440, −7.63655473278594742016856958310, −6.84017025329686856096783340871, −5.45092556221841695561349176061, −5.08311128124940836718387486956, −4.30931575371976229192004754097, −3.6497975724509909167162995833, −2.86793402365969865578982906886, −1.94005044340938649529519534981, −1.36002685170598866063154034074, −0.103649396698279610340775012352,
0.51704255574831223970733447405, 2.03517056884974350120380078216, 2.899485889740140890343805726735, 3.569184019417275170121330753453, 4.66877210357655772570497675587, 5.056348914147028257539999760722, 5.84335692260215208307960153503, 6.861902498022778953569326588201, 7.217443084738802792130701969131, 7.70286249159742576430865473536, 8.54570218660884606303181560428, 9.20530423306822089912784746191, 10.01292122078464571298978950915, 10.56422550948317533155117082832, 11.33686602269269505708061376170, 11.980323573602142651005222236832, 13.15804482954689249751089075823, 13.34929483427759080160838956356, 14.27141996487111536075199632640, 14.969207441425764533828015639761, 15.361032097131062712719268062767, 15.87569120094462772919148211244, 16.63385128211357002150745869193, 17.316294194930854408972141126816, 18.09191436556047983205674281753