L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.766 − 0.642i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.766 − 0.642i)10-s + 11-s + (−0.766 − 0.642i)13-s + (−0.766 + 0.642i)14-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)17-s + (−0.5 + 0.866i)20-s + (−0.173 − 0.984i)22-s + (−0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 + 0.866i)26-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.766 − 0.642i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.766 − 0.642i)10-s + 11-s + (−0.766 − 0.642i)13-s + (−0.766 + 0.642i)14-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)17-s + (−0.5 + 0.866i)20-s + (−0.173 − 0.984i)22-s + (−0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 + 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1371105608 - 1.194028880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1371105608 - 1.194028880i\) |
\(L(1)\) |
\(\approx\) |
\(0.6393102968 - 0.6728668112i\) |
\(L(1)\) |
\(\approx\) |
\(0.6393102968 - 0.6728668112i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.766 - 0.642i)T \) |
| 17 | \( 1 + (0.766 - 0.642i)T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.939 - 0.342i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.939 + 0.342i)T \) |
| 53 | \( 1 + (-0.173 + 0.984i)T \) |
| 59 | \( 1 + (0.939 + 0.342i)T \) |
| 61 | \( 1 + (0.766 + 0.642i)T \) |
| 67 | \( 1 + (-0.173 + 0.984i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + (-0.939 - 0.342i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.939 - 0.342i)T \) |
| 97 | \( 1 + (-0.173 - 0.984i)T \) |
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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
−27.67892712889635700679970043894, −26.58594201213155817707824241093, −25.74708936219432275912125211088, −25.09288804083943237516107883642, −24.231849525368238701568183230358, −22.981178795858531026411726157487, −21.99479616770331233470691375422, −21.63443751038426395509545729286, −19.613460748555073098092245818901, −18.825616266871955967219508559555, −17.917533928878881526899051398961, −16.965058353894686313068332294919, −16.07587738890389654539017175643, −14.64688297015437031761051430331, −14.43069930787484830783107057383, −13.03926123044760635655122350011, −11.88773494282077540145121856844, −10.13059329355618917628834903906, −9.47834932725644476958070912391, −8.39382689047880919684945377947, −6.86096600741273437978204344459, −6.24902403923661638923569643934, −5.1427475210217366939302510777, −3.52038051257271350732807072423, −1.80336550388124948931119736654,
0.46885708348924032847186094471, 1.66792103245550994046201798419, 3.19823354780954580346253649110, 4.409386480369194422490221394805, 5.63249590127930586112079259595, 7.27057256629005343927204746278, 8.662812555886292958371370752832, 9.78221911132118227425098595045, 10.23526579722915817014044514271, 11.800549356491895647885087879526, 12.61780136480796928956939122654, 13.65990887811379090837730849084, 14.35396349935185937360681969110, 16.3186916314003166934599686890, 17.16649454746183127562460481318, 17.853841556028871001989227698334, 19.27834838776472367333971747933, 20.04363140120461278816334895030, 20.74463159076052598521407406089, 21.90672676447094774431101780530, 22.58682633172710729057661456411, 23.74266591417688582741859071578, 25.02874898019382101776658368753, 25.89891118575572053889085961775, 27.10075223731750823620845282537