L(s) = 1 | + 3·3-s + 3·5-s − 2·7-s + 3·9-s + 9·15-s − 6·17-s + 4·19-s − 6·21-s + 4·23-s + 5·25-s − 8·29-s + 7·31-s − 6·35-s + 37-s + 4·41-s − 24·43-s + 9·45-s + 8·47-s + 7·49-s − 18·51-s − 2·53-s + 12·57-s + 59-s + 4·61-s − 6·63-s − 20·67-s + 12·69-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 1.34·5-s − 0.755·7-s + 9-s + 2.32·15-s − 1.45·17-s + 0.917·19-s − 1.30·21-s + 0.834·23-s + 25-s − 1.48·29-s + 1.25·31-s − 1.01·35-s + 0.164·37-s + 0.624·41-s − 3.65·43-s + 1.34·45-s + 1.16·47-s + 49-s − 2.52·51-s − 0.274·53-s + 1.58·57-s + 0.130·59-s + 0.512·61-s − 0.755·63-s − 2.44·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.758205295\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.758205295\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 11 | | \( 1 \) |
good | 3 | $C_4\times C_2$ | \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - 3 T + 4 T^{2} + 3 T^{3} - 29 T^{4} + 3 p T^{5} + 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 20 T^{3} - 19 T^{4} - 20 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 6 T + 19 T^{2} + 12 T^{3} - 251 T^{4} + 12 p T^{5} + 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 4 T - 3 T^{2} + 88 T^{3} - 295 T^{4} + 88 p T^{5} - 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 + 8 T + 35 T^{2} + 48 T^{3} - 631 T^{4} + 48 p T^{5} + 35 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 7 T + 18 T^{2} + 91 T^{3} - 1195 T^{4} + 91 p T^{5} + 18 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - T - 36 T^{2} + 73 T^{3} + 1259 T^{4} + 73 p T^{5} - 36 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 4 T - 25 T^{2} + 264 T^{3} - 31 T^{4} + 264 p T^{5} - 25 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - 8 T + 17 T^{2} + 240 T^{3} - 2719 T^{4} + 240 p T^{5} + 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 2 T - 49 T^{2} - 204 T^{3} + 2189 T^{4} - 204 p T^{5} - 49 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - T - 58 T^{2} + 117 T^{3} + 3305 T^{4} + 117 p T^{5} - 58 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - 4 T - 45 T^{2} + 424 T^{3} + 1049 T^{4} + 424 p T^{5} - 45 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 + 3 T - 62 T^{2} - 399 T^{3} + 3205 T^{4} - 399 p T^{5} - 62 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - 16 T + 183 T^{2} - 1760 T^{3} + 14801 T^{4} - 1760 p T^{5} + 183 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 2 T - 75 T^{2} + 308 T^{3} + 5309 T^{4} + 308 p T^{5} - 75 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 + 2 T - 79 T^{2} - 324 T^{3} + 5909 T^{4} - 324 p T^{5} - 79 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 1015 p T^{5} - 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.42106685749779866455368211720, −6.85254356050670262244959977138, −6.58483533942736339458593338776, −6.56688749907891484618169596009, −6.47447277571424752090507794830, −6.05749693755850727237219074065, −5.81998825288188653518772211395, −5.69264575702249144944728666576, −5.33821784782231695172201129040, −5.01060340559596666817224087118, −4.87850922695875882622487909157, −4.55404179485391781129437347519, −4.49165149735128573401100900865, −3.99122592489209156290938489255, −3.62043204207395836256277456756, −3.33939569761694825677763136207, −3.21485037047170936623488307515, −3.19031435216265834566637521027, −2.68868216983682582746808199439, −2.37836488353728299615303695665, −2.03830571311696940907735680193, −1.93732632722267207459751567449, −1.74198883001627299499788086606, −0.851165717581207586838881437225, −0.63561363980084285658945317226,
0.63561363980084285658945317226, 0.851165717581207586838881437225, 1.74198883001627299499788086606, 1.93732632722267207459751567449, 2.03830571311696940907735680193, 2.37836488353728299615303695665, 2.68868216983682582746808199439, 3.19031435216265834566637521027, 3.21485037047170936623488307515, 3.33939569761694825677763136207, 3.62043204207395836256277456756, 3.99122592489209156290938489255, 4.49165149735128573401100900865, 4.55404179485391781129437347519, 4.87850922695875882622487909157, 5.01060340559596666817224087118, 5.33821784782231695172201129040, 5.69264575702249144944728666576, 5.81998825288188653518772211395, 6.05749693755850727237219074065, 6.47447277571424752090507794830, 6.56688749907891484618169596009, 6.58483533942736339458593338776, 6.85254356050670262244959977138, 7.42106685749779866455368211720