| L(s) = 1 | − 2·4-s + 6·5-s − 3·9-s + 4·16-s + 6·17-s − 12·20-s + 18·25-s + 6·36-s − 18·45-s − 14·49-s − 8·64-s − 12·68-s + 32·73-s + 24·80-s + 9·81-s + 36·85-s − 36·100-s − 18·101-s + 30·125-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s − 18·153-s + ⋯ |
| L(s) = 1 | − 4-s + 2.68·5-s − 9-s + 16-s + 1.45·17-s − 2.68·20-s + 18/5·25-s + 36-s − 2.68·45-s − 2·49-s − 64-s − 1.45·68-s + 3.74·73-s + 2.68·80-s + 81-s + 3.90·85-s − 3.59·100-s − 1.79·101-s + 2.68·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s − 1.45·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.694893148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.694893148\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.829928851074064300694073290188, −9.522270723583321581110893602240, −9.390405948325931390902961357510, −8.632525427372709916279133885770, −8.190291291726257374238102485803, −7.63111517523178875529663610255, −6.54531723660951830540584042474, −6.24844990551280608868718687484, −5.66597647096355167889411363013, −5.23748899123154290652561725026, −4.98465432672254545310032712635, −3.74373759461110443084008273966, −3.01132187067917648663499447218, −2.20948109293983073958098160529, −1.28047961343305408068664694122,
1.28047961343305408068664694122, 2.20948109293983073958098160529, 3.01132187067917648663499447218, 3.74373759461110443084008273966, 4.98465432672254545310032712635, 5.23748899123154290652561725026, 5.66597647096355167889411363013, 6.24844990551280608868718687484, 6.54531723660951830540584042474, 7.63111517523178875529663610255, 8.190291291726257374238102485803, 8.632525427372709916279133885770, 9.390405948325931390902961357510, 9.522270723583321581110893602240, 9.829928851074064300694073290188
