| L(s) = 1 | + 2-s − 9·3-s − 3·4-s + 15·5-s − 9·6-s + 6·7-s − 7·8-s + 54·9-s + 15·10-s + 33·11-s + 27·12-s − 20·13-s + 6·14-s − 135·15-s − 27·16-s + 32·17-s + 54·18-s + 116·19-s − 45·20-s − 54·21-s + 33·22-s + 240·23-s + 63·24-s + 150·25-s − 20·26-s − 270·27-s − 18·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.73·3-s − 3/8·4-s + 1.34·5-s − 0.612·6-s + 0.323·7-s − 0.309·8-s + 2·9-s + 0.474·10-s + 0.904·11-s + 0.649·12-s − 0.426·13-s + 0.114·14-s − 2.32·15-s − 0.421·16-s + 0.456·17-s + 0.707·18-s + 1.40·19-s − 0.503·20-s − 0.561·21-s + 0.319·22-s + 2.17·23-s + 0.535·24-s + 6/5·25-s − 0.150·26-s − 1.92·27-s − 0.121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.155632856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.155632856\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + p^{2} T^{2} + p^{5} T^{4} - p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 6 T + 761 T^{2} - 4820 T^{3} + 761 p^{3} T^{4} - 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 20 T + 1671 T^{2} + 752 p T^{3} + 1671 p^{3} T^{4} + 20 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 32 T + 12059 T^{2} - 336856 T^{3} + 12059 p^{3} T^{4} - 32 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 116 T + 23933 T^{2} - 1591368 T^{3} + 23933 p^{3} T^{4} - 116 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 240 T + 46229 T^{2} - 5659936 T^{3} + 46229 p^{3} T^{4} - 240 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 238 T + 78303 T^{2} - 11180748 T^{3} + 78303 p^{3} T^{4} - 238 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 92 T + 35821 T^{2} + 1288120 T^{3} + 35821 p^{3} T^{4} - 92 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 90 T + 56259 T^{2} + 2753372 T^{3} + 56259 p^{3} T^{4} + 90 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 46 T + 197627 T^{2} + 6094844 T^{3} + 197627 p^{3} T^{4} + 46 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 134 T + 220453 T^{2} + 18926516 T^{3} + 220453 p^{3} T^{4} + 134 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 220 T + 176989 T^{2} + 56662344 T^{3} + 176989 p^{3} T^{4} + 220 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 798 T + 553379 T^{2} + 220344724 T^{3} + 553379 p^{3} T^{4} + 798 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 1236 T + 1040201 T^{2} - 540620792 T^{3} + 1040201 p^{3} T^{4} - 1236 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 342 T + 612635 T^{2} - 157910180 T^{3} + 612635 p^{3} T^{4} - 342 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 764 T + 722241 T^{2} - 305880680 T^{3} + 722241 p^{3} T^{4} - 764 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 1816 T + 2129845 T^{2} - 1498091472 T^{3} + 2129845 p^{3} T^{4} - 1816 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 100 T + 1093435 T^{2} - 82936384 T^{3} + 1093435 p^{3} T^{4} - 100 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 96 T + 666905 T^{2} - 72496384 T^{3} + 666905 p^{3} T^{4} + 96 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 858 T + 587277 T^{2} - 439050316 T^{3} + 587277 p^{3} T^{4} - 858 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 838 T + 615287 T^{2} - 488514452 T^{3} + 615287 p^{3} T^{4} - 838 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1322 T + 2288799 T^{2} + 2058506156 T^{3} + 2288799 p^{3} T^{4} + 1322 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.07904444892634687709549286026, −10.88259697363829445524401388231, −10.38926805398512018628733245544, −9.909103159308129748900987866630, −9.684556257828984750693164202204, −9.549849553089363393120341412171, −9.282261317456584000562431960811, −8.487024843523878192796846856601, −8.396144790289119131559845953362, −7.83122693648406139570091848270, −7.17114604733085893981143726309, −6.79471294058941706190637239887, −6.63572952843573568707216839022, −6.39080057052814645048120108073, −5.76476596052275293442869129411, −5.44902853802903441730155701058, −4.91558040455083219916512788620, −4.79802128578515615762823476461, −4.73248873960455429108959037937, −3.50125993267377666422717976530, −3.40080168087615840203684801713, −2.49641627590220521141409036631, −1.74409730451185268088544075927, −1.01696290428988740337621192414, −0.75331159580547719371251200662,
0.75331159580547719371251200662, 1.01696290428988740337621192414, 1.74409730451185268088544075927, 2.49641627590220521141409036631, 3.40080168087615840203684801713, 3.50125993267377666422717976530, 4.73248873960455429108959037937, 4.79802128578515615762823476461, 4.91558040455083219916512788620, 5.44902853802903441730155701058, 5.76476596052275293442869129411, 6.39080057052814645048120108073, 6.63572952843573568707216839022, 6.79471294058941706190637239887, 7.17114604733085893981143726309, 7.83122693648406139570091848270, 8.396144790289119131559845953362, 8.487024843523878192796846856601, 9.282261317456584000562431960811, 9.549849553089363393120341412171, 9.684556257828984750693164202204, 9.909103159308129748900987866630, 10.38926805398512018628733245544, 10.88259697363829445524401388231, 11.07904444892634687709549286026
