| L(s) = 1 | + 3-s − 4-s − 2·9-s − 12-s − 3·16-s − 3·17-s + 12·23-s + 2·25-s − 5·27-s − 6·29-s + 2·36-s − 2·43-s − 3·48-s − 4·49-s − 3·51-s − 9·53-s − 2·61-s + 7·64-s + 3·68-s + 12·69-s + 2·75-s − 17·79-s + 81-s − 6·87-s − 12·92-s − 2·100-s − 24·101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 2/3·9-s − 0.288·12-s − 3/4·16-s − 0.727·17-s + 2.50·23-s + 2/5·25-s − 0.962·27-s − 1.11·29-s + 1/3·36-s − 0.304·43-s − 0.433·48-s − 4/7·49-s − 0.420·51-s − 1.23·53-s − 0.256·61-s + 7/8·64-s + 0.363·68-s + 1.44·69-s + 0.230·75-s − 1.91·79-s + 1/9·81-s − 0.643·87-s − 1.25·92-s − 1/5·100-s − 2.38·101-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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
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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
−8.814005939386991854258673397734, −8.315362367090927246042100460798, −7.893693096301574074185695347616, −7.26554441247473338444102297178, −6.77954353356762467324058559818, −6.50815932881900973140596153642, −5.53996890439937722528011810590, −5.35482174753973693459348474825, −4.61391027984957382923616551128, −4.26873242206288260778295151385, −3.40320507406871119679961424090, −2.97496837972366575400395370492, −2.36119275940204257676803348358, −1.40937486296158484204981566055, 0,
1.40937486296158484204981566055, 2.36119275940204257676803348358, 2.97496837972366575400395370492, 3.40320507406871119679961424090, 4.26873242206288260778295151385, 4.61391027984957382923616551128, 5.35482174753973693459348474825, 5.53996890439937722528011810590, 6.50815932881900973140596153642, 6.77954353356762467324058559818, 7.26554441247473338444102297178, 7.893693096301574074185695347616, 8.315362367090927246042100460798, 8.814005939386991854258673397734
