| L(s) = 1 | + 6·2-s − 5·4-s − 180·8-s − 555·16-s − 828·17-s + 608·19-s + 600·23-s + 478·31-s + 2.03e3·32-s − 4.96e3·34-s + 3.64e3·38-s + 3.60e3·46-s − 4.33e3·47-s + 4.44e3·49-s − 3.18e3·53-s − 632·61-s + 2.86e3·62-s + 1.89e4·64-s + 4.14e3·68-s − 3.04e3·76-s + 2.09e4·79-s + 2.52e4·83-s − 3.00e3·92-s − 2.59e4·94-s + 2.66e4·98-s − 1.91e4·106-s + 1.02e3·107-s + ⋯ |
| L(s) = 1 | + 3/2·2-s − 0.312·4-s − 2.81·8-s − 2.16·16-s − 2.86·17-s + 1.68·19-s + 1.13·23-s + 0.497·31-s + 1.98·32-s − 4.29·34-s + 2.52·38-s + 1.70·46-s − 1.96·47-s + 1.84·49-s − 1.13·53-s − 0.169·61-s + 0.746·62-s + 4.62·64-s + 0.895·68-s − 0.526·76-s + 3.34·79-s + 3.66·83-s − 0.354·92-s − 2.94·94-s + 2.77·98-s − 1.70·106-s + 0.0896·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.782748975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.782748975\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - 3 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 4441 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 14153 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 34082 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 414 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 16 p T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 300 T + p^{4} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 954878 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 239 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2338 p^{2} T^{2} + p^{8} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 5599538 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 5873278 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2166 T + p^{4} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 1593 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 15696638 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 316 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 18939358 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 47518238 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 47609521 T^{2} + p^{8} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10450 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12633 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 76456478 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 134587273 T^{2} + p^{8} T^{4} \) |
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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
−10.07170849903350519232545446223, −9.345718570807899249536933071897, −9.262449874989938251825030416548, −9.139225282107012079173891217929, −8.318750056112559511531789804147, −8.173072083196391158003779548138, −7.36461434097534877319682821051, −6.78056204638243053838726511838, −6.38240100436307254072755977538, −6.10866149330715570789987416362, −5.24510192972590971729369197138, −5.06056392149586406666056878322, −4.69448249148793969122384676170, −4.26666875491485835832034767543, −3.58825513782979243998986641994, −3.30816371112308986368997291850, −2.64082826713242301691006409211, −2.08434433349122113068114062805, −0.914842229111898404844347337849, −0.38869769384841650663271258554,
0.38869769384841650663271258554, 0.914842229111898404844347337849, 2.08434433349122113068114062805, 2.64082826713242301691006409211, 3.30816371112308986368997291850, 3.58825513782979243998986641994, 4.26666875491485835832034767543, 4.69448249148793969122384676170, 5.06056392149586406666056878322, 5.24510192972590971729369197138, 6.10866149330715570789987416362, 6.38240100436307254072755977538, 6.78056204638243053838726511838, 7.36461434097534877319682821051, 8.173072083196391158003779548138, 8.318750056112559511531789804147, 9.139225282107012079173891217929, 9.262449874989938251825030416548, 9.345718570807899249536933071897, 10.07170849903350519232545446223
