| L(s) = 1 | − 8·7-s − 6·9-s + 4·17-s + 8·23-s + 25-s − 16·31-s − 12·41-s + 8·47-s + 34·49-s + 48·63-s − 12·73-s + 27·81-s − 12·89-s − 28·97-s + 8·103-s + 36·113-s − 32·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s − 64·161-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 2·9-s + 0.970·17-s + 1.66·23-s + 1/5·25-s − 2.87·31-s − 1.87·41-s + 1.16·47-s + 34/7·49-s + 6.04·63-s − 1.40·73-s + 3·81-s − 1.27·89-s − 2.84·97-s + 0.788·103-s + 3.38·113-s − 2.93·119-s − 0.545·121-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 − 1.94·153-s + 0.0798·157-s − 5.04·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−11.01108015186625578670651780817, −10.39655063289957855259191225662, −9.851840183892789826086741161768, −9.206637608661514184739644068270, −9.022035638415746178025932660186, −8.421702762202697929017375581976, −7.28963827504896405934712065825, −6.97238544391138654785050907556, −6.22671236180701133336548488807, −5.70093888866808737514223874840, −5.28966562607622137666656232279, −3.64413717916439174649981742998, −3.26180587408034592610487247010, −2.75667504176180688667939525456, 0,
2.75667504176180688667939525456, 3.26180587408034592610487247010, 3.64413717916439174649981742998, 5.28966562607622137666656232279, 5.70093888866808737514223874840, 6.22671236180701133336548488807, 6.97238544391138654785050907556, 7.28963827504896405934712065825, 8.421702762202697929017375581976, 9.022035638415746178025932660186, 9.206637608661514184739644068270, 9.851840183892789826086741161768, 10.39655063289957855259191225662, 11.01108015186625578670651780817
