L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)6-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)12-s + (0.309 + 0.951i)13-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s − 18-s + (0.809 − 0.587i)19-s + (0.809 + 0.587i)22-s + (−0.309 + 0.951i)23-s − 24-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)6-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)12-s + (0.309 + 0.951i)13-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s − 18-s + (0.809 − 0.587i)19-s + (0.809 + 0.587i)22-s + (−0.309 + 0.951i)23-s − 24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2633672097 - 0.3183562684i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2633672097 - 0.3183562684i\) |
\(L(1)\) |
\(\approx\) |
\(0.5824586302 + 0.07358163672i\) |
\(L(1)\) |
\(\approx\) |
\(0.5824586302 + 0.07358163672i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
| 13 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (-0.809 + 0.587i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
| 23 | \( 1 + (-0.309 + 0.951i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.309 - 0.951i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + (0.809 - 0.587i)T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (-0.309 + 0.951i)T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−27.62547821030372105436635377502, −26.895284039965064672659021185860, −25.908662785898974762543890964328, −24.59077778633803591820297896793, −22.97805903137641475485755245416, −22.65479665659713350774063748929, −21.71421365786638479807195185056, −20.4776137696191380322252059245, −20.15092049292703614034786202986, −18.43823341011719794495838974731, −17.8796036967544705510590739158, −16.9214422120256411204923982342, −15.84763104379424290387309026638, −14.626463772274298496061502045215, −13.16535076938226390196431283061, −12.19951555822167555923580742113, −11.37901772605068381907806834311, −10.29082340675444936397989852022, −9.6501114124724011079290358605, −8.36717299525120114081330225810, −6.85509176433739185941978823273, −5.26026729686465745434857705138, −4.319468661144668659062883328036, −3.05433055834011106768682364467, −1.30282980084778761275041082378,
0.20666164734757702458229909577, 1.632254133193134758000219341350, 4.049784880111157905833759738346, 5.40477644401512397253507605180, 6.291751811431024074458170512645, 7.179551486133372981706602848062, 8.34804936850924506513205675746, 9.460726502424340102111721141698, 10.87727980895107831165347961597, 11.77286130596428868111533194495, 13.36979338884881695638529666968, 13.81949049647821090241713459241, 15.359259917325772563080609295225, 16.306143524532776788783004906590, 17.0899598156237416030912673575, 17.96169473459338355381002737924, 18.90340626061705062340858250780, 19.62675752939924302618582809025, 21.572004940286450461740286605448, 22.33695163676947493414696257329, 23.37878545662764105943615981716, 24.22724081403906481166050816688, 24.65283505500680131191047637648, 26.03716002903915602182322517117, 26.77440812732645011362259820485