L(s) = 1 | + (0.406 − 0.913i)2-s + (−0.743 + 0.669i)3-s + (−0.669 − 0.743i)4-s + (0.309 + 0.951i)6-s + (−0.951 + 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (0.994 + 0.104i)12-s + (0.587 − 0.809i)13-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.866 − 0.5i)18-s + (−0.669 + 0.743i)19-s + (−0.951 − 0.309i)22-s + (−0.406 + 0.913i)23-s + (0.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.406 − 0.913i)2-s + (−0.743 + 0.669i)3-s + (−0.669 − 0.743i)4-s + (0.309 + 0.951i)6-s + (−0.951 + 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (0.994 + 0.104i)12-s + (0.587 − 0.809i)13-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.866 − 0.5i)18-s + (−0.669 + 0.743i)19-s + (−0.951 − 0.309i)22-s + (−0.406 + 0.913i)23-s + (0.5 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3992460186 + 0.3304677676i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3992460186 + 0.3304677676i\) |
\(L(1)\) |
\(\approx\) |
\(0.7489038794 - 0.2031958430i\) |
\(L(1)\) |
\(\approx\) |
\(0.7489038794 - 0.2031958430i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.406 - 0.913i)T \) |
| 3 | \( 1 + (-0.743 + 0.669i)T \) |
| 11 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (0.587 - 0.809i)T \) |
| 17 | \( 1 + (-0.207 + 0.978i)T \) |
| 19 | \( 1 + (-0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.406 + 0.913i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.978 - 0.207i)T \) |
| 37 | \( 1 + (0.994 + 0.104i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.207 + 0.978i)T \) |
| 53 | \( 1 + (-0.743 + 0.669i)T \) |
| 59 | \( 1 + (-0.913 + 0.406i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (-0.207 + 0.978i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.994 - 0.104i)T \) |
| 79 | \( 1 + (0.978 - 0.207i)T \) |
| 83 | \( 1 + (-0.951 + 0.309i)T \) |
| 89 | \( 1 + (-0.913 - 0.406i)T \) |
| 97 | \( 1 + (-0.951 - 0.309i)T \) |
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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
−26.84979428876926058597261515576, −25.72703621799631791360295874660, −24.990555119498017957371623942735, −23.96150945610319457759309959733, −23.31933263968854612427049726036, −22.501927011988799159800213123035, −21.59470487310237245626067070115, −20.28189828412627092057542284533, −18.70930552367435150010934678406, −18.07345810011658611761145741058, −17.081264523887049674331650738442, −16.278518623519957725252688881633, −15.22581949002911180873967768762, −13.9902047070260166673328397626, −13.114412732062881650357145930953, −12.20544027939462577284059457302, −11.17662090518755357683339856207, −9.55794222333282776643302660620, −8.25520076683178398614580723305, −7.07323376735794772432924220502, −6.46831967549059132920100907044, −5.16680750234381032320682298446, −4.231793014229701290400790221145, −2.2247670193618861857594794353, −0.18928703330587124837297749785,
1.32809150676557367978701509294, 3.26020447188779860934641224811, 4.09499836811286741095326783984, 5.5229343984319062520579564918, 6.12149064624250935149683500293, 8.322594921888685890997712019240, 9.49028286470654254750411119050, 10.683721730436244363798733360, 11.07086266680593343504505102355, 12.34778162653997889864849912341, 13.21439459382012531503656866815, 14.53945898433841261599062512819, 15.48037785621112433024472241933, 16.63712941441340774549725645546, 17.784442375777429013969157032586, 18.68162947032440671204000151203, 19.8640085037921595137458476038, 20.867938828284609885560800475876, 21.68145551086934497752420879156, 22.32455599560017906973984701311, 23.458132212377948719026359736263, 23.96740872746370377569288042831, 25.6516128878224614197732957518, 26.933944252647383569401135388509, 27.59288213162954188626837846148