| L(s) = 1 | − 2·5-s + 3·9-s − 6·13-s − 2·17-s + 3·25-s − 18·29-s + 4·37-s − 8·41-s − 6·45-s + 49-s + 8·53-s − 16·61-s + 12·65-s + 4·73-s + 4·85-s + 8·89-s − 26·97-s + 12·101-s − 6·109-s + 28·113-s − 18·117-s + 3·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 9-s − 1.66·13-s − 0.485·17-s + 3/5·25-s − 3.34·29-s + 0.657·37-s − 1.24·41-s − 0.894·45-s + 1/7·49-s + 1.09·53-s − 2.04·61-s + 1.48·65-s + 0.468·73-s + 0.433·85-s + 0.847·89-s − 2.63·97-s + 1.19·101-s − 0.574·109-s + 2.63·113-s − 1.66·117-s + 3/11·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−9.716558810176534364860056345490, −9.025895505037624522078550270792, −8.548341637730933961960631631377, −7.73731776455161087713955528993, −7.33986466138726521878856236101, −7.32545514222645839389554035710, −6.55070763080365592509044517141, −5.79502306005565665039377390440, −5.11248277682454093810005074849, −4.64056350631119746271470996824, −4.00451609950028562939589775427, −3.51475005035302395747602864778, −2.51163217316890307126702952381, −1.70936955426510995854988214866, 0,
1.70936955426510995854988214866, 2.51163217316890307126702952381, 3.51475005035302395747602864778, 4.00451609950028562939589775427, 4.64056350631119746271470996824, 5.11248277682454093810005074849, 5.79502306005565665039377390440, 6.55070763080365592509044517141, 7.32545514222645839389554035710, 7.33986466138726521878856236101, 7.73731776455161087713955528993, 8.548341637730933961960631631377, 9.025895505037624522078550270792, 9.716558810176534364860056345490
