| L(s) = 1 | + 4-s − 2·7-s − 3·16-s + 2·25-s − 2·28-s − 6·31-s − 4·37-s + 2·43-s − 3·49-s − 12·61-s − 7·64-s + 4·67-s + 73-s + 4·79-s − 12·97-s + 2·100-s − 18·103-s − 4·109-s + 6·112-s − 2·121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 3/4·16-s + 2/5·25-s − 0.377·28-s − 1.07·31-s − 0.657·37-s + 0.304·43-s − 3/7·49-s − 1.53·61-s − 7/8·64-s + 0.488·67-s + 0.117·73-s + 0.450·79-s − 1.21·97-s + 1/5·100-s − 1.77·103-s − 0.383·109-s + 0.566·112-s − 0.181·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.328·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289737 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289737 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.629083932375796096800797260811, −8.183029188681408827165644430257, −7.56068658384449826906220734072, −7.18126769922215475671340318066, −6.67632316476880693523526584236, −6.40038811554054155194942404547, −5.77596435001089733645172495640, −5.27779615193093456793302363658, −4.67849391427767565028381587599, −4.05555799868875227571268644430, −3.44905392934496712050613025301, −2.87968961024498799673346089621, −2.24477395196645851609933650161, −1.43339146186309693888282332176, 0,
1.43339146186309693888282332176, 2.24477395196645851609933650161, 2.87968961024498799673346089621, 3.44905392934496712050613025301, 4.05555799868875227571268644430, 4.67849391427767565028381587599, 5.27779615193093456793302363658, 5.77596435001089733645172495640, 6.40038811554054155194942404547, 6.67632316476880693523526584236, 7.18126769922215475671340318066, 7.56068658384449826906220734072, 8.183029188681408827165644430257, 8.629083932375796096800797260811
