| L(s) = 1 | − 5-s + 2·13-s − 2·17-s + 25-s + 4·29-s + 2·37-s − 16·41-s − 2·49-s − 18·53-s + 4·61-s − 2·65-s + 10·73-s + 2·85-s − 12·89-s − 14·97-s + 28·101-s − 12·109-s + 22·113-s − 10·121-s − 125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s − 0.485·17-s + 1/5·25-s + 0.742·29-s + 0.328·37-s − 2.49·41-s − 2/7·49-s − 2.47·53-s + 0.512·61-s − 0.248·65-s + 1.17·73-s + 0.216·85-s − 1.27·89-s − 1.42·97-s + 2.78·101-s − 1.14·109-s + 2.06·113-s − 0.909·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.332·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.138138702535008216732275109050, −7.84130140456261647118993604057, −7.16612365519815919159760674332, −6.80903459029275937111683730698, −6.34495779718512174830532378297, −6.01603839414729323195236723411, −5.23211503907463651270953390821, −4.86199820676129201534527707620, −4.44415684055343815268568302002, −3.70136543603565082576265709272, −3.37478385709343679302840344549, −2.72875102374585163969860361574, −1.93565686763223844396763774548, −1.21000763333816743285504562586, 0,
1.21000763333816743285504562586, 1.93565686763223844396763774548, 2.72875102374585163969860361574, 3.37478385709343679302840344549, 3.70136543603565082576265709272, 4.44415684055343815268568302002, 4.86199820676129201534527707620, 5.23211503907463651270953390821, 6.01603839414729323195236723411, 6.34495779718512174830532378297, 6.80903459029275937111683730698, 7.16612365519815919159760674332, 7.84130140456261647118993604057, 8.138138702535008216732275109050
