| L(s) = 1 | + 2·5-s − 4·9-s − 4·13-s + 12·17-s + 3·25-s − 8·29-s + 4·37-s + 16·41-s − 8·45-s + 4·49-s − 4·53-s − 28·61-s − 8·65-s + 12·73-s + 7·81-s + 24·85-s + 12·89-s + 20·97-s + 12·109-s − 20·113-s + 16·117-s + 10·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 4/3·9-s − 1.10·13-s + 2.91·17-s + 3/5·25-s − 1.48·29-s + 0.657·37-s + 2.49·41-s − 1.19·45-s + 4/7·49-s − 0.549·53-s − 3.58·61-s − 0.992·65-s + 1.40·73-s + 7/9·81-s + 2.60·85-s + 1.27·89-s + 2.03·97-s + 1.14·109-s − 1.88·113-s + 1.47·117-s + 0.909·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.206934032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.206934032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 84 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−9.761164842599230452270933274350, −9.417813960759149650388470402767, −9.146917843984336564199460927881, −8.934543866025143240598883397938, −7.965800338953623128029847401656, −7.84804635143844064735293788203, −7.60923321559064389382275585544, −7.16521819535438178940961557869, −6.26415304508219311984899254644, −6.11974118255946195368307126920, −5.58415167840856839644784371779, −5.49936600934448183561518176378, −4.90639145286840195823956743878, −4.40058944195331033734735040382, −3.58082100235383699110499102578, −3.20684118995595838914477826137, −2.72144630647887323038483034788, −2.22259305792153845108702480225, −1.45358908317161389346517381853, −0.64110331270867126694107758633,
0.64110331270867126694107758633, 1.45358908317161389346517381853, 2.22259305792153845108702480225, 2.72144630647887323038483034788, 3.20684118995595838914477826137, 3.58082100235383699110499102578, 4.40058944195331033734735040382, 4.90639145286840195823956743878, 5.49936600934448183561518176378, 5.58415167840856839644784371779, 6.11974118255946195368307126920, 6.26415304508219311984899254644, 7.16521819535438178940961557869, 7.60923321559064389382275585544, 7.84804635143844064735293788203, 7.965800338953623128029847401656, 8.934543866025143240598883397938, 9.146917843984336564199460927881, 9.417813960759149650388470402767, 9.761164842599230452270933274350
