L(s) = 1 | + (−0.654 − 0.755i)3-s + (−0.909 − 0.415i)7-s + (−0.142 + 0.989i)9-s + (−0.281 + 0.959i)11-s + (0.415 + 0.909i)13-s + (−0.540 + 0.841i)17-s + (−0.540 − 0.841i)19-s + (0.281 + 0.959i)21-s + (0.841 − 0.540i)27-s + (−0.540 + 0.841i)29-s + (−0.654 + 0.755i)31-s + (0.909 − 0.415i)33-s + (0.142 − 0.989i)37-s + (0.415 − 0.909i)39-s + (0.142 + 0.989i)41-s + ⋯ |
L(s) = 1 | + (−0.654 − 0.755i)3-s + (−0.909 − 0.415i)7-s + (−0.142 + 0.989i)9-s + (−0.281 + 0.959i)11-s + (0.415 + 0.909i)13-s + (−0.540 + 0.841i)17-s + (−0.540 − 0.841i)19-s + (0.281 + 0.959i)21-s + (0.841 − 0.540i)27-s + (−0.540 + 0.841i)29-s + (−0.654 + 0.755i)31-s + (0.909 − 0.415i)33-s + (0.142 − 0.989i)37-s + (0.415 − 0.909i)39-s + (0.142 + 0.989i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1827694041 - 0.3671286535i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1827694041 - 0.3671286535i\) |
\(L(1)\) |
\(\approx\) |
\(0.6330761500 - 0.1152799179i\) |
\(L(1)\) |
\(\approx\) |
\(0.6330761500 - 0.1152799179i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (-0.909 - 0.415i)T \) |
| 11 | \( 1 + (-0.281 + 0.959i)T \) |
| 13 | \( 1 + (0.415 + 0.909i)T \) |
| 17 | \( 1 + (-0.540 + 0.841i)T \) |
| 19 | \( 1 + (-0.540 - 0.841i)T \) |
| 29 | \( 1 + (-0.540 + 0.841i)T \) |
| 31 | \( 1 + (-0.654 + 0.755i)T \) |
| 37 | \( 1 + (0.142 - 0.989i)T \) |
| 41 | \( 1 + (0.142 + 0.989i)T \) |
| 43 | \( 1 + (-0.654 - 0.755i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.415 - 0.909i)T \) |
| 59 | \( 1 + (0.909 - 0.415i)T \) |
| 61 | \( 1 + (-0.755 - 0.654i)T \) |
| 67 | \( 1 + (-0.959 + 0.281i)T \) |
| 71 | \( 1 + (0.959 - 0.281i)T \) |
| 73 | \( 1 + (-0.540 - 0.841i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (0.142 - 0.989i)T \) |
| 89 | \( 1 + (0.654 + 0.755i)T \) |
| 97 | \( 1 + (0.989 - 0.142i)T \) |
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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
−20.58444400263602380144060363837, −19.73414773256974014180930086277, −18.711167628398018597644026839046, −18.34432503595413283896387128410, −17.31428529345921945874551605826, −16.5986094453273031889823593595, −16.02449286854129841517556464005, −15.435366518213347898966878150894, −14.75944308713579231962557600014, −13.57388184868211486262796829668, −13.003014562615851645260744920624, −12.11193917454531958864462014229, −11.37194940643530498100769676853, −10.65048974534007476980188565567, −9.96830705779235666227121253036, −9.20124410823910691177795592395, −8.490610471313399968915215973021, −7.455531831021992299665984476885, −6.18243605171098410794012897779, −5.98094722185835262156181181301, −5.10442742683433957566774199734, −4.02329235302846664099337044076, −3.316776131104502367170924865919, −2.5083037715568017701439401591, −0.85225178238095438613191962465,
0.201119658337919093080124582122, 1.613011386479027795959875304670, 2.23920391162991906883441101590, 3.52363965440808911898357288623, 4.43353948227545866832268876312, 5.28892075055707042585449557171, 6.34493652808978211409187497499, 6.83143304888954759441764209887, 7.39335397347553522156928845116, 8.539977832214975533669769233850, 9.300210916648410239404425173408, 10.33748353677054907611543864806, 10.88482427518234440420660713025, 11.72517817900618690268948762837, 12.68786399069474130681686485800, 12.979560830210103051898955976748, 13.72225698058840749335837485169, 14.70220094109628650612306521488, 15.58423933525752474478925050405, 16.45674849978308044722134856801, 16.87230027899568190911019454853, 17.83262144081406989341790932961, 18.24803945607780563247861108743, 19.28552278498731172036894824819, 19.64232043995826261849974404976