| L(s) = 1 | + 9-s − 4·13-s + 2·17-s − 8·19-s − 6·25-s + 8·43-s − 14·49-s − 4·53-s + 8·59-s − 8·67-s + 81-s − 8·83-s − 12·89-s − 36·101-s + 32·103-s − 4·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 1.10·13-s + 0.485·17-s − 1.83·19-s − 6/5·25-s + 1.21·43-s − 2·49-s − 0.549·53-s + 1.04·59-s − 0.977·67-s + 1/9·81-s − 0.878·83-s − 1.27·89-s − 3.58·101-s + 3.15·103-s − 0.369·117-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 + 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.063016264776882028484939879353, −8.355230373829598766062247754786, −8.098990694093691505068092710868, −7.48026403891189149604247587328, −7.14328906397907053732568751561, −6.42897107072744719896577758454, −6.12988314631940998265391004819, −5.43016342758798685600037411651, −4.89441641834110651309625753413, −4.25303028692796488061253338187, −3.90806135734356722472911536103, −2.96100787019000394786081708512, −2.31953951885724301169050983275, −1.57698730169915032251498272270, 0,
1.57698730169915032251498272270, 2.31953951885724301169050983275, 2.96100787019000394786081708512, 3.90806135734356722472911536103, 4.25303028692796488061253338187, 4.89441641834110651309625753413, 5.43016342758798685600037411651, 6.12988314631940998265391004819, 6.42897107072744719896577758454, 7.14328906397907053732568751561, 7.48026403891189149604247587328, 8.098990694093691505068092710868, 8.355230373829598766062247754786, 9.063016264776882028484939879353
