| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s − 3·16-s + 7·25-s + 4·27-s + 36-s − 16·43-s + 6·48-s − 14·49-s − 2·61-s − 7·64-s − 14·75-s − 8·79-s − 11·81-s + 7·100-s + 20·103-s + 4·108-s + 22·121-s + 127-s + 32·129-s + 131-s + 137-s + 139-s − 3·144-s + 28·147-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s − 0.577·12-s − 3/4·16-s + 7/5·25-s + 0.769·27-s + 1/6·36-s − 2.43·43-s + 0.866·48-s − 2·49-s − 0.256·61-s − 7/8·64-s − 1.61·75-s − 0.900·79-s − 1.22·81-s + 7/10·100-s + 1.97·103-s + 0.384·108-s + 2·121-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + 2.30·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
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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
−8.548705411131615187333245735619, −8.413924923181321157661018721715, −7.58435050934765083191540699545, −7.12986318506267137737340144223, −6.64809577269047253502795016781, −6.37045452395473691530551678679, −5.91475950324760340741955262076, −5.15284448233542809174349730392, −4.87931349212911460694179701039, −4.43976161925919705270752829969, −3.42578145921934543050776786950, −2.99544918153476500911441200311, −2.11523865706184741264380506451, −1.27248056496288501941272854048, 0,
1.27248056496288501941272854048, 2.11523865706184741264380506451, 2.99544918153476500911441200311, 3.42578145921934543050776786950, 4.43976161925919705270752829969, 4.87931349212911460694179701039, 5.15284448233542809174349730392, 5.91475950324760340741955262076, 6.37045452395473691530551678679, 6.64809577269047253502795016781, 7.12986318506267137737340144223, 7.58435050934765083191540699545, 8.413924923181321157661018721715, 8.548705411131615187333245735619
