| L(s) = 1 | + 8·5-s + 9-s − 4·13-s − 4·17-s + 38·25-s + 4·37-s + 12·41-s + 8·45-s − 10·49-s − 28·61-s − 32·65-s − 20·73-s + 81-s − 32·85-s − 28·89-s + 20·97-s + 12·109-s + 4·113-s − 4·117-s − 6·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 1/3·9-s − 1.10·13-s − 0.970·17-s + 38/5·25-s + 0.657·37-s + 1.87·41-s + 1.19·45-s − 1.42·49-s − 3.58·61-s − 3.96·65-s − 2.34·73-s + 1/9·81-s − 3.47·85-s − 2.96·89-s + 2.03·97-s + 1.14·109-s + 0.376·113-s − 0.369·117-s − 0.545·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.082273503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.082273503\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.433707596930861266358550422356, −8.985911500488081519481984584802, −8.743300071949026671666068984095, −7.69025257463999263217129645276, −7.27371486188202795087403560747, −6.52923673370585434887828168703, −6.30170361535159036673173480205, −5.81778647559970487461786585347, −5.46781891072947205774781799219, −4.60632905768716122167488407077, −4.59332442534898507060628341827, −2.92708427914109016867182117304, −2.63009703571906191631419054606, −1.91249343134731656186986110897, −1.43950309180346771366044691508,
1.43950309180346771366044691508, 1.91249343134731656186986110897, 2.63009703571906191631419054606, 2.92708427914109016867182117304, 4.59332442534898507060628341827, 4.60632905768716122167488407077, 5.46781891072947205774781799219, 5.81778647559970487461786585347, 6.30170361535159036673173480205, 6.52923673370585434887828168703, 7.27371486188202795087403560747, 7.69025257463999263217129645276, 8.743300071949026671666068984095, 8.985911500488081519481984584802, 9.433707596930861266358550422356
