| L(s) = 1 | + 3-s − 4-s + 2·7-s − 12-s − 3·16-s + 6·17-s + 4·19-s + 2·21-s + 6·23-s − 5·25-s − 4·27-s − 2·28-s + 8·37-s − 3·48-s − 2·49-s + 6·51-s + 4·57-s − 12·59-s + 7·64-s − 6·68-s + 6·69-s − 16·73-s − 5·75-s − 4·76-s − 7·81-s − 2·84-s − 6·92-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s + 0.755·7-s − 0.288·12-s − 3/4·16-s + 1.45·17-s + 0.917·19-s + 0.436·21-s + 1.25·23-s − 25-s − 0.769·27-s − 0.377·28-s + 1.31·37-s − 0.433·48-s − 2/7·49-s + 0.840·51-s + 0.529·57-s − 1.56·59-s + 7/8·64-s − 0.727·68-s + 0.722·69-s − 1.87·73-s − 0.577·75-s − 0.458·76-s − 7/9·81-s − 0.218·84-s − 0.625·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.365607129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.365607129\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.93276644227757333031625645199, −10.10163779476938448780305838903, −9.547209602513455319226040582624, −9.318548112113446454658579589047, −8.630232138722726726261809434406, −8.061094800920401536075916654311, −7.60164427339101666472115139255, −7.19352417367746344744532317129, −6.17942759152891363259097027725, −5.55040724880780680584074602961, −4.93466888383200436229205489342, −4.28171211549881236314270799644, −3.44639679545555833386417403331, −2.68632856176424194301388983736, −1.42513955094109757627168452969,
1.42513955094109757627168452969, 2.68632856176424194301388983736, 3.44639679545555833386417403331, 4.28171211549881236314270799644, 4.93466888383200436229205489342, 5.55040724880780680584074602961, 6.17942759152891363259097027725, 7.19352417367746344744532317129, 7.60164427339101666472115139255, 8.061094800920401536075916654311, 8.630232138722726726261809434406, 9.318548112113446454658579589047, 9.547209602513455319226040582624, 10.10163779476938448780305838903, 10.93276644227757333031625645199
