| L(s) = 1 | + 4-s + 16·16-s − 50·25-s − 76·37-s − 232·43-s + 47·64-s + 236·67-s + 188·79-s − 50·100-s + 212·109-s − 206·121-s + 127-s + 131-s + 137-s + 139-s − 76·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 676·169-s − 232·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1/4·4-s + 16-s − 2·25-s − 2.05·37-s − 5.39·43-s + 0.734·64-s + 3.52·67-s + 2.37·79-s − 1/2·100-s + 1.94·109-s − 1.70·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.513·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4·169-s − 1.34·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.269250123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.269250123\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 15 T^{4} - p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 206 T^{2} + 27795 T^{4} + 206 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 734 T^{2} + 258915 T^{4} + 734 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 1234 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 38 T + 75 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 5582 T^{2} + 23268243 T^{4} + 5582 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T + 9435 T^{2} - 118 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 2914 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T + 2595 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
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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
−8.015951762243454572759871615146, −7.66739127771355078333662029884, −7.34758407639092734836276284482, −6.95600389503612017138057559358, −6.75345456614683950384569944341, −6.61989234596792268078699810932, −6.52071123271210110278614824388, −6.13916784173033926181912764516, −5.70953176912478281101878264568, −5.45075012683790756626766985531, −5.37274017123747871648680192085, −4.96762005255159533924116055482, −4.95481608732772090725236387253, −4.49090188408965088973457271170, −3.98047196954975433734718693878, −3.67714383981345678608745891938, −3.65355677724699831252209863324, −3.25891549352790073616987521593, −3.07540317483051559883104221787, −2.50478270095404445312009561233, −1.90266493672753517797654689876, −1.80808869549421209016989844524, −1.68825100429735420124709611504, −0.76674291266331167511519419060, −0.34821576803977595239447722160,
0.34821576803977595239447722160, 0.76674291266331167511519419060, 1.68825100429735420124709611504, 1.80808869549421209016989844524, 1.90266493672753517797654689876, 2.50478270095404445312009561233, 3.07540317483051559883104221787, 3.25891549352790073616987521593, 3.65355677724699831252209863324, 3.67714383981345678608745891938, 3.98047196954975433734718693878, 4.49090188408965088973457271170, 4.95481608732772090725236387253, 4.96762005255159533924116055482, 5.37274017123747871648680192085, 5.45075012683790756626766985531, 5.70953176912478281101878264568, 6.13916784173033926181912764516, 6.52071123271210110278614824388, 6.61989234596792268078699810932, 6.75345456614683950384569944341, 6.95600389503612017138057559358, 7.34758407639092734836276284482, 7.66739127771355078333662029884, 8.015951762243454572759871615146
