L(s) = 1 | + (0.989 − 0.143i)2-s + (−0.467 + 0.884i)3-s + (0.959 − 0.283i)4-s + (0.986 + 0.164i)5-s + (−0.336 + 0.941i)6-s + (0.875 − 0.483i)7-s + (0.908 − 0.417i)8-s + (−0.563 − 0.826i)9-s + (0.999 + 0.0212i)10-s + (−0.158 − 0.987i)11-s + (−0.198 + 0.980i)12-s + (0.145 + 0.989i)13-s + (0.796 − 0.604i)14-s + (−0.606 + 0.795i)15-s + (0.839 − 0.543i)16-s + (0.412 + 0.910i)17-s + ⋯ |
L(s) = 1 | + (0.989 − 0.143i)2-s + (−0.467 + 0.884i)3-s + (0.959 − 0.283i)4-s + (0.986 + 0.164i)5-s + (−0.336 + 0.941i)6-s + (0.875 − 0.483i)7-s + (0.908 − 0.417i)8-s + (−0.563 − 0.826i)9-s + (0.999 + 0.0212i)10-s + (−0.158 − 0.987i)11-s + (−0.198 + 0.980i)12-s + (0.145 + 0.989i)13-s + (0.796 − 0.604i)14-s + (−0.606 + 0.795i)15-s + (0.839 − 0.543i)16-s + (0.412 + 0.910i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.515106062 + 0.9360115431i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.515106062 + 0.9360115431i\) |
\(L(1)\) |
\(\approx\) |
\(2.314767183 + 0.3032841209i\) |
\(L(1)\) |
\(\approx\) |
\(2.314767183 + 0.3032841209i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4729 | \( 1 \) |
good | 2 | \( 1 + (0.989 - 0.143i)T \) |
| 3 | \( 1 + (-0.467 + 0.884i)T \) |
| 5 | \( 1 + (0.986 + 0.164i)T \) |
| 7 | \( 1 + (0.875 - 0.483i)T \) |
| 11 | \( 1 + (-0.158 - 0.987i)T \) |
| 13 | \( 1 + (0.145 + 0.989i)T \) |
| 17 | \( 1 + (0.412 + 0.910i)T \) |
| 19 | \( 1 + (0.361 + 0.932i)T \) |
| 23 | \( 1 + (-0.748 - 0.663i)T \) |
| 29 | \( 1 + (0.143 - 0.989i)T \) |
| 31 | \( 1 + (0.930 + 0.366i)T \) |
| 37 | \( 1 + (0.877 + 0.479i)T \) |
| 41 | \( 1 + (-0.0318 + 0.999i)T \) |
| 43 | \( 1 + (-0.137 + 0.990i)T \) |
| 47 | \( 1 + (0.100 - 0.994i)T \) |
| 53 | \( 1 + (-0.820 - 0.571i)T \) |
| 59 | \( 1 + (-0.187 - 0.982i)T \) |
| 61 | \( 1 + (0.0531 + 0.998i)T \) |
| 67 | \( 1 + (-0.679 - 0.733i)T \) |
| 71 | \( 1 + (0.0451 + 0.998i)T \) |
| 73 | \( 1 + (-0.558 - 0.829i)T \) |
| 79 | \( 1 + (-0.866 + 0.5i)T \) |
| 83 | \( 1 + (-0.966 + 0.257i)T \) |
| 89 | \( 1 + (-0.397 - 0.917i)T \) |
| 97 | \( 1 + (-0.875 + 0.483i)T \) |
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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
−17.92575171020306770064144853976, −17.52315249036834407776597061575, −16.97141597929236927358923926316, −15.91455223473781379360938223660, −15.44674646855339862136289306492, −14.47710844812934892554423065339, −14.02695631596193301696221026323, −13.40040534839595445532982245544, −12.7558831421323702335616934256, −12.208942579677637161732496685914, −11.62909148629091687806536728464, −10.86177488445974758010449310475, −10.20676453024966852609888807021, −9.21632313023678647541526693189, −8.25198702566178016371316158369, −7.48662272645337310192604114379, −7.10436381014948705993119260052, −6.05922678548688911861342513070, −5.59439489660487199933159929409, −5.04260031894743545651607663033, −4.48497633675788218568863471259, −2.9550049638874037262421513281, −2.49426630871785707539100642334, −1.73171656090632111855620874331, −1.03611402699688208842146042477,
1.08906887143046814075011904887, 1.752511924483215325040060984494, 2.77678739464560755177049336659, 3.55009125254083866133287096890, 4.35430058324117053102697011388, 4.77808119759946880395629087699, 5.825082549100885253054753852503, 6.03014235797599177365041026048, 6.731212842714560169763276237286, 7.98072201219609110856135797359, 8.551254608760463669959738943227, 9.92170626817844428263653177865, 10.01815845003268479090410726484, 10.879227964795275881359412430323, 11.436690271766940935836315859682, 11.946153559766042896719884040409, 12.92698438937486330285986341953, 13.72724765871697411538912667272, 14.25999617605411468743541778660, 14.57813296624642375912076909492, 15.39582493011740527994795651998, 16.43047426931681213667436373510, 16.57967462330571097041727991924, 17.263566244861450753933536645403, 18.14501657197179218458750281750