L(s) = 1 | + (0.986 + 0.164i)3-s + (−0.879 + 0.475i)5-s + (0.401 + 0.915i)7-s + (0.945 + 0.324i)9-s + (0.0825 + 0.996i)11-s + (−0.879 + 0.475i)13-s + (−0.945 + 0.324i)15-s + (−0.879 + 0.475i)17-s + (0.879 + 0.475i)19-s + (0.245 + 0.969i)21-s + (0.677 + 0.735i)23-s + (0.546 − 0.837i)25-s + (0.879 + 0.475i)27-s + (−0.401 − 0.915i)29-s + (0.0825 + 0.996i)31-s + ⋯ |
L(s) = 1 | + (0.986 + 0.164i)3-s + (−0.879 + 0.475i)5-s + (0.401 + 0.915i)7-s + (0.945 + 0.324i)9-s + (0.0825 + 0.996i)11-s + (−0.879 + 0.475i)13-s + (−0.945 + 0.324i)15-s + (−0.879 + 0.475i)17-s + (0.879 + 0.475i)19-s + (0.245 + 0.969i)21-s + (0.677 + 0.735i)23-s + (0.546 − 0.837i)25-s + (0.879 + 0.475i)27-s + (−0.401 − 0.915i)29-s + (0.0825 + 0.996i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09405595847 + 1.966178211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09405595847 + 1.966178211i\) |
\(L(1)\) |
\(\approx\) |
\(1.097862881 + 0.6115271032i\) |
\(L(1)\) |
\(\approx\) |
\(1.097862881 + 0.6115271032i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 229 | \( 1 \) |
good | 3 | \( 1 + (0.986 + 0.164i)T \) |
| 5 | \( 1 + (-0.879 + 0.475i)T \) |
| 7 | \( 1 + (0.401 + 0.915i)T \) |
| 11 | \( 1 + (0.0825 + 0.996i)T \) |
| 13 | \( 1 + (-0.879 + 0.475i)T \) |
| 17 | \( 1 + (-0.879 + 0.475i)T \) |
| 19 | \( 1 + (0.879 + 0.475i)T \) |
| 23 | \( 1 + (0.677 + 0.735i)T \) |
| 29 | \( 1 + (-0.401 - 0.915i)T \) |
| 31 | \( 1 + (0.0825 + 0.996i)T \) |
| 37 | \( 1 + (0.245 + 0.969i)T \) |
| 41 | \( 1 + (0.945 - 0.324i)T \) |
| 43 | \( 1 + (-0.245 - 0.969i)T \) |
| 47 | \( 1 + (-0.945 - 0.324i)T \) |
| 53 | \( 1 + (-0.986 - 0.164i)T \) |
| 59 | \( 1 + (-0.245 + 0.969i)T \) |
| 61 | \( 1 + (0.945 + 0.324i)T \) |
| 67 | \( 1 + (-0.945 - 0.324i)T \) |
| 71 | \( 1 + (0.0825 - 0.996i)T \) |
| 73 | \( 1 + (0.546 - 0.837i)T \) |
| 79 | \( 1 + (0.401 - 0.915i)T \) |
| 83 | \( 1 + (-0.245 + 0.969i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.546 + 0.837i)T \) |
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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
−21.04770123769955709386729819400, −20.2780286067561442548123063299, −19.89226356927151186222851763370, −19.201528481661303824041576460942, −18.29317870600584493340097925007, −17.34879867671897490714888696218, −16.34735288259440882788636353414, −15.80175091173421481728909612875, −14.73047514628109191819331633174, −14.25205270510884981234048628829, −13.179264709142466924925315109947, −12.77180890360325789812934766723, −11.45963558965890191244274124305, −10.93651722915313124688611143849, −9.66143610208788079764094006359, −8.93743001402432828599550271070, −8.02183241711980570085516266718, −7.51552652153673548085476312903, −6.699631075424791578006034060645, −5.073978324476046273032746695263, −4.3654457099987691281617503304, −3.438366086559964252195384831526, −2.62024568289073248337126362145, −1.1509458341115307928555906836, −0.38041256108008667046325459893,
1.63931232519982377644676823294, 2.447889538337454438294650480726, 3.36013087590563109343406645758, 4.368135573571255435247203677758, 5.06337978777729783446067026156, 6.60513319592873535221540787993, 7.43369661352155348319906256071, 8.03911493114107378851612086038, 9.025706154116562579723643925461, 9.6672169431917851163951672823, 10.68027775969375575708679889190, 11.77228417802117746112596996197, 12.267472634373457642509484929170, 13.33028940885075027787743870738, 14.364715340261046231897985618164, 15.0404529839966180603507862514, 15.34653142813206928083414203275, 16.23619492867089010474819038537, 17.51749971394992126247267945525, 18.27872854005363881959614869956, 19.16238870105725990863339807627, 19.5862723357124312105077177574, 20.43404272511399230884634528208, 21.22347299509456351556373497278, 22.08543895076779907509622809064