L(s) = 1 | + 2-s − 2·5-s − 7-s − 2·10-s − 13-s − 14-s − 16-s − 14·23-s − 3·25-s − 26-s − 4·29-s + 15·31-s + 32-s + 2·35-s − 3·37-s + 4·41-s − 3·43-s − 14·46-s + 18·47-s − 11·49-s − 3·50-s + 6·53-s − 4·58-s + 13·61-s + 15·62-s − 2·64-s + 2·65-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.894·5-s − 0.377·7-s − 0.632·10-s − 0.277·13-s − 0.267·14-s − 1/4·16-s − 2.91·23-s − 3/5·25-s − 0.196·26-s − 0.742·29-s + 2.69·31-s + 0.176·32-s + 0.338·35-s − 0.493·37-s + 0.624·41-s − 0.457·43-s − 2.06·46-s + 2.62·47-s − 1.57·49-s − 0.424·50-s + 0.824·53-s − 0.525·58-s + 1.66·61-s + 1.90·62-s − 1/4·64-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.800786107\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.800786107\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 19 | | \( 1 \) |
good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} - T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 8 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 12 T^{2} + 17 T^{3} + 12 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 36 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + T + 18 T^{2} + 29 T^{3} + 18 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 + 14 T + 97 T^{2} + 488 T^{3} + 97 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 55 T^{2} + 136 T^{3} + 55 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 15 T + 144 T^{2} - 29 p T^{3} + 144 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 91 T^{2} - 232 T^{3} + 91 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 3 T + 108 T^{2} + 271 T^{3} + 108 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 87 T^{2} - 312 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 153 T^{2} - 36 T^{3} + 153 p T^{4} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 194 T^{2} - 1513 T^{3} + 194 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 9 T + 120 T^{2} - 665 T^{3} + 120 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 18 T + 225 T^{2} - 1908 T^{3} + 225 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 19 T + 302 T^{2} - 2743 T^{3} + 302 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 11 T + 240 T^{2} - 1567 T^{3} + 240 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 169 T^{2} - 472 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 16 T + 343 T^{2} - 2956 T^{3} + 343 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 23 T^{2} - 1060 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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
−7.71435956057497221456898792166, −7.58001348248452133358347317486, −7.01428097868081438510683021992, −6.84342452620837481203659925822, −6.71592137314820929266741566390, −6.28262575387051116509695506992, −6.24442470622530667956531931773, −5.79815328535240558271846677929, −5.65370879349837794886391780141, −5.39826874295739582424784674223, −4.87562099503737659412609585312, −4.78693825267277723977174248434, −4.61771212231854216260714124870, −3.94720159198705705690894644285, −3.92867131592538724401952333760, −3.92510201337969948468173452061, −3.56771318262696213402071862467, −3.21822089476621381776739753226, −2.58908043888323618132161169445, −2.44929385036094861160676087479, −2.16246149303960155526409308322, −1.95717406589227275335648745645, −1.15722349165664516667868881151, −0.78156480231356303898854791481, −0.29627147163659246408247627009,
0.29627147163659246408247627009, 0.78156480231356303898854791481, 1.15722349165664516667868881151, 1.95717406589227275335648745645, 2.16246149303960155526409308322, 2.44929385036094861160676087479, 2.58908043888323618132161169445, 3.21822089476621381776739753226, 3.56771318262696213402071862467, 3.92510201337969948468173452061, 3.92867131592538724401952333760, 3.94720159198705705690894644285, 4.61771212231854216260714124870, 4.78693825267277723977174248434, 4.87562099503737659412609585312, 5.39826874295739582424784674223, 5.65370879349837794886391780141, 5.79815328535240558271846677929, 6.24442470622530667956531931773, 6.28262575387051116509695506992, 6.71592137314820929266741566390, 6.84342452620837481203659925822, 7.01428097868081438510683021992, 7.58001348248452133358347317486, 7.71435956057497221456898792166