| L(s) = 1 | + 3-s + 4-s − 6·5-s + 5·7-s + 6·11-s + 12-s − 5·13-s − 6·15-s − 3·16-s − 12·17-s + 3·19-s − 6·20-s + 5·21-s + 12·23-s + 19·25-s − 27-s + 5·28-s − 6·29-s + 18·31-s + 6·33-s − 30·35-s − 5·39-s + 6·41-s + 43-s + 6·44-s + 18·47-s − 3·48-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 2.68·5-s + 1.88·7-s + 1.80·11-s + 0.288·12-s − 1.38·13-s − 1.54·15-s − 3/4·16-s − 2.91·17-s + 0.688·19-s − 1.34·20-s + 1.09·21-s + 2.50·23-s + 19/5·25-s − 0.192·27-s + 0.944·28-s − 1.11·29-s + 3.23·31-s + 1.04·33-s − 5.07·35-s − 0.800·39-s + 0.937·41-s + 0.152·43-s + 0.904·44-s + 2.62·47-s − 0.433·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.490404490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490404490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 18 T + 155 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 21 T + 244 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.09876794542864125199963168148, −11.65465249673277861787919217237, −11.27465400794026421130014241726, −10.99068506300137973154318278167, −10.74464266318358946982836169655, −9.487173721108784005892269861464, −8.963861951490664014025115954578, −8.751587122598577868940854573561, −8.351419275874168317516292484973, −7.60176701623702954731864721956, −7.43252351844914267691456711688, −6.83115283938602103233905443083, −6.71318095459065128024954931937, −5.27400891547754757199905135924, −4.50514945182106868857248201359, −4.28468374499511648347748618302, −4.14582000475039803770619125736, −2.85137306554832622274177953431, −2.37527920241892230098893977412, −0.981520855423620467473514821937,
0.981520855423620467473514821937, 2.37527920241892230098893977412, 2.85137306554832622274177953431, 4.14582000475039803770619125736, 4.28468374499511648347748618302, 4.50514945182106868857248201359, 5.27400891547754757199905135924, 6.71318095459065128024954931937, 6.83115283938602103233905443083, 7.43252351844914267691456711688, 7.60176701623702954731864721956, 8.351419275874168317516292484973, 8.751587122598577868940854573561, 8.963861951490664014025115954578, 9.487173721108784005892269861464, 10.74464266318358946982836169655, 10.99068506300137973154318278167, 11.27465400794026421130014241726, 11.65465249673277861787919217237, 12.09876794542864125199963168148
