| L(s) = 1 | + 3·2-s + 3-s + 3·4-s + 3·6-s + 9-s − 9·11-s + 3·12-s − 14·13-s − 2·16-s + 6·17-s + 3·18-s + 9·19-s − 27·22-s − 6·23-s + 15·25-s − 42·26-s − 4·27-s + 9·29-s + 6·32-s − 9·33-s + 18·34-s + 3·36-s + 24·37-s + 27·38-s − 14·39-s − 18·41-s − 5·43-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.577·3-s + 3/2·4-s + 1.22·6-s + 1/3·9-s − 2.71·11-s + 0.866·12-s − 3.88·13-s − 1/2·16-s + 1.45·17-s + 0.707·18-s + 2.06·19-s − 5.75·22-s − 1.25·23-s + 3·25-s − 8.23·26-s − 0.769·27-s + 1.67·29-s + 1.06·32-s − 1.56·33-s + 3.08·34-s + 1/2·36-s + 3.94·37-s + 4.37·38-s − 2.24·39-s − 2.81·41-s − 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.307219756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.307219756\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} ) \) |
| 3 | $D_4\times C_2$ | \( 1 - T + 5 T^{3} - 11 T^{4} + 5 p T^{5} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 101 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 9 T + 54 T^{2} + 243 T^{3} + 905 T^{4} + 243 p T^{5} + 54 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 9 T + 56 T^{2} - 261 T^{3} + 993 T^{4} - 261 p T^{5} + 56 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 2 T^{2} - 72 T^{3} - 201 T^{4} - 72 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 9 T + 8 T^{2} - 135 T^{3} + 2139 T^{4} - 135 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 210 T^{2} + 1836 T^{3} + 13151 T^{4} + 1836 p T^{5} + 210 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T - 20 T^{2} - 205 T^{3} - 899 T^{4} - 205 p T^{5} - 20 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 78 T^{2} + 4595 T^{4} - 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T + 21 T^{2} + 54 T^{3} - 2692 T^{4} + 54 p T^{5} + 21 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T + 70 T^{2} - 376 T^{3} + 391 T^{4} - 376 p T^{5} + 70 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 828 T^{3} + 14855 T^{4} - 828 p T^{5} + 150 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 270 T^{2} + 31667 T^{4} - 270 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 33 T + 588 T^{2} - 7425 T^{3} + 75011 T^{4} - 7425 p T^{5} + 588 p^{2} T^{6} - 33 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 39 T + 812 T^{2} - 11895 T^{3} + 132795 T^{4} - 11895 p T^{5} + 812 p^{2} T^{6} - 39 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.74877425168365594260237677889, −7.34150167552812145586146284221, −7.28464483994551031939350551727, −6.98189212705122811679862003759, −6.58528229148462483192431559472, −6.27429343856061941558013860965, −6.16473037672556940125161873214, −5.72595265434186874292787510418, −5.47607828098225386784987627837, −5.24256810482218738934865840794, −4.96206817363485237646196803907, −4.82805757096900479260107752463, −4.74073964159259720591452148693, −4.64977096967864172658224527500, −4.56667528476319610584813868021, −3.74252111441710993804958462428, −3.48497073732612541650986951648, −3.29498762492748357425125107314, −2.84146579155933846267424105971, −2.77811873909924684767672166968, −2.72149990093153078315780972755, −2.21130596328021549322311126868, −1.92137794749078930091982327511, −0.946768860521896181542105793764, −0.55432840540967627536145525108,
0.55432840540967627536145525108, 0.946768860521896181542105793764, 1.92137794749078930091982327511, 2.21130596328021549322311126868, 2.72149990093153078315780972755, 2.77811873909924684767672166968, 2.84146579155933846267424105971, 3.29498762492748357425125107314, 3.48497073732612541650986951648, 3.74252111441710993804958462428, 4.56667528476319610584813868021, 4.64977096967864172658224527500, 4.74073964159259720591452148693, 4.82805757096900479260107752463, 4.96206817363485237646196803907, 5.24256810482218738934865840794, 5.47607828098225386784987627837, 5.72595265434186874292787510418, 6.16473037672556940125161873214, 6.27429343856061941558013860965, 6.58528229148462483192431559472, 6.98189212705122811679862003759, 7.28464483994551031939350551727, 7.34150167552812145586146284221, 7.74877425168365594260237677889
