| L(s) = 1 | + 3·4-s − 6·13-s + 5·16-s − 4·17-s + 16·23-s − 25-s − 4·29-s − 8·43-s + 10·49-s − 18·52-s + 12·53-s + 20·61-s + 3·64-s − 12·68-s − 16·79-s + 48·92-s − 3·100-s + 4·101-s + 32·103-s − 16·107-s + 28·113-s − 12·116-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1.66·13-s + 5/4·16-s − 0.970·17-s + 3.33·23-s − 1/5·25-s − 0.742·29-s − 1.21·43-s + 10/7·49-s − 2.49·52-s + 1.64·53-s + 2.56·61-s + 3/8·64-s − 1.45·68-s − 1.80·79-s + 5.00·92-s − 0.299·100-s + 0.398·101-s + 3.15·103-s − 1.54·107-s + 2.63·113-s − 1.11·116-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.512792485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.512792485\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−11.18908303893844382145638457612, −10.50652447586590134584175538090, −10.11193025431004835574999797372, −9.774418599448382876298292178453, −9.054775305234376381944097009976, −8.827158965474191394829576124834, −8.311914435489103729665529237720, −7.54255789681969772151180481301, −7.06357887030790450688126903654, −7.02100999387547807945134952475, −6.80011609323267747166074356929, −5.75680074107899084556349865968, −5.60460074777884679055400582417, −4.78877822530374383925331334590, −4.57191923479342937387491900822, −3.50186868744688653130295428518, −3.04934857004459317386903042290, −2.32595134924025721943551554109, −2.10172711546210300610380058133, −0.889768438408529013270686134904,
0.889768438408529013270686134904, 2.10172711546210300610380058133, 2.32595134924025721943551554109, 3.04934857004459317386903042290, 3.50186868744688653130295428518, 4.57191923479342937387491900822, 4.78877822530374383925331334590, 5.60460074777884679055400582417, 5.75680074107899084556349865968, 6.80011609323267747166074356929, 7.02100999387547807945134952475, 7.06357887030790450688126903654, 7.54255789681969772151180481301, 8.311914435489103729665529237720, 8.827158965474191394829576124834, 9.054775305234376381944097009976, 9.774418599448382876298292178453, 10.11193025431004835574999797372, 10.50652447586590134584175538090, 11.18908303893844382145638457612
