L(s) = 1 | + 3-s − 4-s + 9-s − 12-s − 6·13-s + 16-s + 6·25-s + 27-s − 36-s − 6·39-s − 8·43-s + 48-s + 49-s + 6·52-s − 12·61-s − 64-s + 6·75-s + 81-s − 6·100-s + 16·103-s − 108-s − 6·117-s + 6·121-s + 127-s − 8·129-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s − 1.66·13-s + 1/4·16-s + 6/5·25-s + 0.192·27-s − 1/6·36-s − 0.960·39-s − 1.21·43-s + 0.144·48-s + 1/7·49-s + 0.832·52-s − 1.53·61-s − 1/8·64-s + 0.692·75-s + 1/9·81-s − 3/5·100-s + 1.57·103-s − 0.0962·108-s − 0.554·117-s + 6/11·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.025688309168167905022269870458, −7.41024547576845205147702456890, −7.29897087109296957472268982667, −6.72513032249390455481214147424, −6.24592091240545789421425441517, −5.63620093622988544848667175177, −5.06857161439399297527291846350, −4.69582305853857169532061766063, −4.44004501872011223263262636066, −3.58903262929681548446051761425, −3.19604100345843133398004727348, −2.58874498006366720886251421530, −2.04062794059719293544223922165, −1.15644919730322647591299718428, 0,
1.15644919730322647591299718428, 2.04062794059719293544223922165, 2.58874498006366720886251421530, 3.19604100345843133398004727348, 3.58903262929681548446051761425, 4.44004501872011223263262636066, 4.69582305853857169532061766063, 5.06857161439399297527291846350, 5.63620093622988544848667175177, 6.24592091240545789421425441517, 6.72513032249390455481214147424, 7.29897087109296957472268982667, 7.41024547576845205147702456890, 8.025688309168167905022269870458