| L(s) = 1 | + 3·2-s + 3-s + 3·4-s + 14·5-s + 3·6-s − 35·7-s + 8-s − 16·9-s + 42·10-s + 16·11-s + 3·12-s + 65·13-s − 105·14-s + 14·15-s − 33·16-s + 29·17-s − 48·18-s − 57·19-s + 42·20-s − 35·21-s + 48·22-s − 101·23-s + 24-s − 108·25-s + 195·26-s − 151·27-s − 105·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.192·3-s + 3/8·4-s + 1.25·5-s + 0.204·6-s − 1.88·7-s + 0.0441·8-s − 0.592·9-s + 1.32·10-s + 0.438·11-s + 0.0721·12-s + 1.38·13-s − 2.00·14-s + 0.240·15-s − 0.515·16-s + 0.413·17-s − 0.628·18-s − 0.688·19-s + 0.469·20-s − 0.363·21-s + 0.465·22-s − 0.915·23-s + 0.00850·24-s − 0.863·25-s + 1.47·26-s − 1.07·27-s − 0.708·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6859 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6859 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.925150084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.925150084\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 19 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 5 p T^{3} + 3 p^{4} T^{4} - 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - T + 17 T^{2} + 118 T^{3} + 17 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 14 T + 304 T^{2} - 3572 T^{3} + 304 p^{3} T^{4} - 14 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 p T + 24 p^{2} T^{2} + 21691 T^{3} + 24 p^{5} T^{4} + 5 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 16 T + 3942 T^{2} - 41410 T^{3} + 3942 p^{3} T^{4} - 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 p T + 7335 T^{2} - 280762 T^{3} + 7335 p^{3} T^{4} - 5 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 29 T + 5514 T^{2} - 503573 T^{3} + 5514 p^{3} T^{4} - 29 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 101 T + 31877 T^{2} + 2079558 T^{3} + 31877 p^{3} T^{4} + 101 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 13 p T + 81935 T^{2} - 13844910 T^{3} + 81935 p^{3} T^{4} - 13 p^{7} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 140 T + 51757 T^{2} + 5897128 T^{3} + 51757 p^{3} T^{4} + 140 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 290 T + 105187 T^{2} + 19377292 T^{3} + 105187 p^{3} T^{4} + 290 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 956 T + 508879 T^{2} - 163355096 T^{3} + 508879 p^{3} T^{4} - 956 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 570 T + 145938 T^{2} + 24674476 T^{3} + 145938 p^{3} T^{4} + 570 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 66 T + 280158 T^{2} - 10764012 T^{3} + 280158 p^{3} T^{4} - 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 817 T + 657711 T^{2} - 260089834 T^{3} + 657711 p^{3} T^{4} - 817 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 265 T + 458145 T^{2} - 77293258 T^{3} + 458145 p^{3} T^{4} - 265 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 988 T + 726644 T^{2} - 371638582 T^{3} + 726644 p^{3} T^{4} - 988 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 207 T + 842361 T^{2} + 117000634 T^{3} + 842361 p^{3} T^{4} + 207 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 846 T + 1246593 T^{2} - 603857484 T^{3} + 1246593 p^{3} T^{4} - 846 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 627 T + 811566 T^{2} - 342245479 T^{3} + 811566 p^{3} T^{4} - 627 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 382 T + 809229 T^{2} - 432705284 T^{3} + 809229 p^{3} T^{4} - 382 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 766 T + 1679713 T^{2} + 797249332 T^{3} + 1679713 p^{3} T^{4} + 766 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 172 T + 1270123 T^{2} + 165585880 T^{3} + 1270123 p^{3} T^{4} + 172 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2450 T + 4122563 T^{2} + 4668536612 T^{3} + 4122563 p^{3} T^{4} + 2450 p^{6} T^{5} + p^{9} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.16888560917391758379492595941, −15.75455184312948749633617729254, −15.72836041583817526130278755856, −14.89072574467196254879323755163, −14.24330505534209114655038697667, −13.94458282194016520156643692652, −13.80453494862963605565654409667, −13.15925565171524995021716161471, −12.98894972556472582823645446400, −12.55946505828843771273162685542, −11.75231523569651038041613208500, −11.53949360615419159706432468369, −10.59158517759838794819515282478, −10.20883657387201471139500452795, −9.527134055457222237238578222502, −9.438289098868931336409521284559, −8.551515017393265439692682089695, −8.093737082628268959245413428410, −6.66970108304130382083214591104, −6.60536515965165763306633666862, −5.80090437546682022466286782719, −5.60155656187740077362057676202, −4.11894757696298139562956381698, −3.58036129084453324196323961651, −2.42435818975861193846168331153,
2.42435818975861193846168331153, 3.58036129084453324196323961651, 4.11894757696298139562956381698, 5.60155656187740077362057676202, 5.80090437546682022466286782719, 6.60536515965165763306633666862, 6.66970108304130382083214591104, 8.093737082628268959245413428410, 8.551515017393265439692682089695, 9.438289098868931336409521284559, 9.527134055457222237238578222502, 10.20883657387201471139500452795, 10.59158517759838794819515282478, 11.53949360615419159706432468369, 11.75231523569651038041613208500, 12.55946505828843771273162685542, 12.98894972556472582823645446400, 13.15925565171524995021716161471, 13.80453494862963605565654409667, 13.94458282194016520156643692652, 14.24330505534209114655038697667, 14.89072574467196254879323755163, 15.72836041583817526130278755856, 15.75455184312948749633617729254, 16.16888560917391758379492595941
