L(s) = 1 | + (−0.841 + 0.540i)3-s + (0.415 + 0.909i)5-s + (0.959 − 0.281i)7-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (−0.959 − 0.281i)13-s + (−0.841 − 0.540i)15-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.654 + 0.755i)21-s + (−0.654 + 0.755i)25-s + (0.142 + 0.989i)27-s + (−0.142 + 0.989i)29-s + (−0.841 − 0.540i)31-s + (−0.959 − 0.281i)33-s + ⋯ |
L(s) = 1 | + (−0.841 + 0.540i)3-s + (0.415 + 0.909i)5-s + (0.959 − 0.281i)7-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (−0.959 − 0.281i)13-s + (−0.841 − 0.540i)15-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (−0.654 + 0.755i)21-s + (−0.654 + 0.755i)25-s + (0.142 + 0.989i)27-s + (−0.142 + 0.989i)29-s + (−0.841 − 0.540i)31-s + (−0.959 − 0.281i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7111909106 + 1.076729550i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7111909106 + 1.076729550i\) |
\(L(1)\) |
\(\approx\) |
\(0.8628975937 + 0.4210481654i\) |
\(L(1)\) |
\(\approx\) |
\(0.8628975937 + 0.4210481654i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.959 - 0.281i)T \) |
| 11 | \( 1 + (0.654 + 0.755i)T \) |
| 13 | \( 1 + (-0.959 - 0.281i)T \) |
| 17 | \( 1 + (-0.142 + 0.989i)T \) |
| 19 | \( 1 + (0.142 + 0.989i)T \) |
| 29 | \( 1 + (-0.142 + 0.989i)T \) |
| 31 | \( 1 + (-0.841 - 0.540i)T \) |
| 37 | \( 1 + (0.415 - 0.909i)T \) |
| 41 | \( 1 + (0.415 + 0.909i)T \) |
| 43 | \( 1 + (-0.841 + 0.540i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.959 + 0.281i)T \) |
| 59 | \( 1 + (0.959 + 0.281i)T \) |
| 61 | \( 1 + (0.841 + 0.540i)T \) |
| 67 | \( 1 + (0.654 - 0.755i)T \) |
| 71 | \( 1 + (0.654 - 0.755i)T \) |
| 73 | \( 1 + (-0.142 - 0.989i)T \) |
| 79 | \( 1 + (0.959 + 0.281i)T \) |
| 83 | \( 1 + (-0.415 + 0.909i)T \) |
| 89 | \( 1 + (0.841 - 0.540i)T \) |
| 97 | \( 1 + (0.415 + 0.909i)T \) |
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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
−29.58060805391190405893378701819, −28.8162605501141968180309984924, −27.7908425781104068294759989373, −26.994458306924141334277006690530, −25.07662948554512841904379044696, −24.42363049952314867664858846788, −23.79339508778249633081551960055, −22.21280000119547995127120368485, −21.501599054232700947390957466169, −20.17991571993627594194857762817, −18.92245498673701864873288691373, −17.682005704791671388645428867022, −17.05902208392097799704809704469, −15.97344549447729242779332008316, −14.26585914033969274248888660519, −13.21848910966914142698458696989, −11.94523330617818750317027361140, −11.29912879065779390543300740327, −9.569958817076489571476168376237, −8.31103128943902432729595089774, −6.91484084776923333973023220680, −5.46248851137550244832340106086, −4.68120245092933230636497717886, −2.07388394478194674423780893993, −0.675585747098607964481219342732,
1.74326981650460929127055213301, 3.80991530204000556504147132120, 5.11601592361277779822024887575, 6.40186722886582272152100549114, 7.61646625995718237255854537494, 9.569721712563948454136586678601, 10.49730682232510423790731945202, 11.43622514152918811879244828132, 12.64306980516852390151225968573, 14.6214660379216703616916134108, 14.84037501363039116968190860418, 16.649986022342650632012977709538, 17.54301399030236973431003812620, 18.23978053377389612595622251616, 19.86703926308540094621746252671, 21.161186460963174116872997545360, 22.02622087888949005983536434520, 22.85219433629511835448746630118, 23.94842494662172116922687374776, 25.208727876736604418699565770561, 26.54819565859175912219021329954, 27.27248269180289795591194532442, 28.21938888434475341821211739431, 29.5256242921364533651667027065, 30.1525683146824678227463085157