| L(s) = 1 | + 5-s − 6·9-s − 4·13-s + 25-s − 16·31-s + 12·37-s − 12·41-s − 16·43-s − 6·45-s + 2·49-s + 12·53-s − 4·65-s + 16·67-s + 27·81-s − 32·83-s − 12·89-s + 24·117-s − 6·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·155-s + 157-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 2·9-s − 1.10·13-s + 1/5·25-s − 2.87·31-s + 1.97·37-s − 1.87·41-s − 2.43·43-s − 0.894·45-s + 2/7·49-s + 1.64·53-s − 0.496·65-s + 1.95·67-s + 3·81-s − 3.51·83-s − 1.27·89-s + 2.21·117-s − 0.545·121-s + 0.0894·125-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 − 1.28·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.851840183892789826086741161768, −9.022035638415746178025932660186, −8.722962238907988148168144658858, −8.235526990771594438753697507946, −7.66639207779752266719368903542, −6.97238544391138654785050907556, −6.59517541927070476620564206344, −5.70093888866808737514223874840, −5.45551464583715971420860983354, −5.04480103462109494033371671985, −4.03115530852322444125343230912, −3.26180587408034592610487247010, −2.64709909765349774471756798901, −1.89767369121488302583154708212, 0,
1.89767369121488302583154708212, 2.64709909765349774471756798901, 3.26180587408034592610487247010, 4.03115530852322444125343230912, 5.04480103462109494033371671985, 5.45551464583715971420860983354, 5.70093888866808737514223874840, 6.59517541927070476620564206344, 6.97238544391138654785050907556, 7.66639207779752266719368903542, 8.235526990771594438753697507946, 8.722962238907988148168144658858, 9.022035638415746178025932660186, 9.851840183892789826086741161768
