| L(s) = 1 | + 4·3-s − 4·7-s + 10·9-s + 16-s − 4·19-s − 16·21-s + 32·27-s − 40·31-s − 4·37-s + 4·48-s + 8·49-s − 16·57-s − 32·61-s − 40·63-s + 20·67-s + 8·73-s − 80·79-s + 89·81-s − 160·93-s − 28·97-s + 8·109-s − 16·111-s − 4·112-s + 127-s + 131-s + 16·133-s + 137-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.51·7-s + 10/3·9-s + 1/4·16-s − 0.917·19-s − 3.49·21-s + 6.15·27-s − 7.18·31-s − 0.657·37-s + 0.577·48-s + 8/7·49-s − 2.11·57-s − 4.09·61-s − 5.03·63-s + 2.44·67-s + 0.936·73-s − 9.00·79-s + 89/9·81-s − 16.5·93-s − 2.84·97-s + 0.766·109-s − 1.51·111-s − 0.377·112-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1568212066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1568212066\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T^{4} - 15 T^{8} - p^{4} T^{12} + p^{8} T^{16} \) |
| 5 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2}( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 + 2 T + 2 T^{2} - 24 T^{3} - 73 T^{4} - 24 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 206 T^{4} + 27795 T^{8} + 206 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 2 T + 2 T^{2} - 72 T^{3} - 433 T^{4} - 72 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 26 T^{2} + 147 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 50 T^{2} + 1659 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 2 T + 2 T^{2} - 144 T^{3} - 1513 T^{4} - 144 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 2722 T^{4} + 4583523 T^{8} - 2722 p^{4} T^{12} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 50 T^{2} + 651 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 1666 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 - 3442 T^{4} - 269997 T^{8} - 3442 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 10 T + 50 T^{2} + 840 T^{3} - 8689 T^{4} + 840 p T^{5} + 50 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 5794 T^{4} + 8158755 T^{8} - 5794 p^{4} T^{12} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 10 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 3374 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 15518 T^{4} + 178066083 T^{8} + 15518 p^{4} T^{12} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 14 T + 98 T^{2} - 1344 T^{3} - 18817 T^{4} - 1344 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.76189096386381570070946140902, −4.63864725332895118203073143125, −4.44487677358451694820468708561, −4.13653072911740632463002757049, −4.06306649563595096426307907961, −4.06075463510290048440173933567, −3.94772449596502164374740914076, −3.87162640086216596868383671015, −3.46565069730993514588416873934, −3.43083137562711586876908852936, −3.29195401144391934787623422629, −3.12178148521797688312611986393, −3.04927052367034177292258821531, −2.91565235915600801063385555965, −2.80286239047113522615924923616, −2.61488503387479758038166917235, −2.31292254736253142222025339398, −2.02140599346058453468075572188, −2.00562687841878007749009153089, −1.81845394087668786338812573208, −1.51354470790136810273353816060, −1.48453424146428443847951494766, −1.29659564951024181329522911681, −0.62386480223666033888601524920, −0.05535976487160293539049137089,
0.05535976487160293539049137089, 0.62386480223666033888601524920, 1.29659564951024181329522911681, 1.48453424146428443847951494766, 1.51354470790136810273353816060, 1.81845394087668786338812573208, 2.00562687841878007749009153089, 2.02140599346058453468075572188, 2.31292254736253142222025339398, 2.61488503387479758038166917235, 2.80286239047113522615924923616, 2.91565235915600801063385555965, 3.04927052367034177292258821531, 3.12178148521797688312611986393, 3.29195401144391934787623422629, 3.43083137562711586876908852936, 3.46565069730993514588416873934, 3.87162640086216596868383671015, 3.94772449596502164374740914076, 4.06075463510290048440173933567, 4.06306649563595096426307907961, 4.13653072911740632463002757049, 4.44487677358451694820468708561, 4.63864725332895118203073143125, 4.76189096386381570070946140902
Plot not available for L-functions of degree greater than 10.
