| L(s) = 1 | − 2·5-s − 6·9-s + 4·13-s + 4·17-s + 3·25-s + 4·29-s − 12·37-s − 12·41-s + 12·45-s + 2·49-s − 12·53-s + 4·61-s − 8·65-s − 12·73-s + 27·81-s − 8·85-s − 12·89-s − 28·97-s − 12·101-s − 28·109-s + 36·113-s − 24·117-s − 6·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2·9-s + 1.10·13-s + 0.970·17-s + 3/5·25-s + 0.742·29-s − 1.97·37-s − 1.87·41-s + 1.78·45-s + 2/7·49-s − 1.64·53-s + 0.512·61-s − 0.992·65-s − 1.40·73-s + 3·81-s − 0.867·85-s − 1.27·89-s − 2.84·97-s − 1.19·101-s − 2.68·109-s + 3.38·113-s − 2.21·117-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−9.206637608661514184739644068270, −8.438761280447847026288494531448, −8.421702762202697929017375581976, −8.187189248591461729679142296814, −7.28963827504896405934712065825, −6.81982863783073968912914119025, −6.22671236180701133336548488807, −5.61529786752672156928392151688, −5.28966562607622137666656232279, −4.55607707806255331316875895933, −3.64413717916439174649981742998, −3.31415635628336508949565358363, −2.75667504176180688667939525456, −1.46465509347525672726427035614, 0,
1.46465509347525672726427035614, 2.75667504176180688667939525456, 3.31415635628336508949565358363, 3.64413717916439174649981742998, 4.55607707806255331316875895933, 5.28966562607622137666656232279, 5.61529786752672156928392151688, 6.22671236180701133336548488807, 6.81982863783073968912914119025, 7.28963827504896405934712065825, 8.187189248591461729679142296814, 8.421702762202697929017375581976, 8.438761280447847026288494531448, 9.206637608661514184739644068270
